TPTP Problem File: ITP210^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP210^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00353_010205
% 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_00353_010205 [Des22]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.67 v8.1.0
% Syntax : Number of formulae : 9631 (3108 unt;1008 typ; 0 def)
% Number of atoms : 26171 (9472 equ; 5 cnn)
% Maximal formula atoms : 48 ( 3 avg)
% Number of connectives : 191215 (2361 ~; 258 |;1768 &;175425 @)
% ( 0 <=>;11403 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 8723 (8723 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1001 ( 997 usr; 21 con; 0-10 aty)
% Number of variables : 33599 (5062 ^;26789 !; 639 ?;33599 :)
% (1109 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 14:35:50.718
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
thf(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Old__Datatype_Onode,type,
old_node: $tType > $tType > $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
% Explicit typings (985)
thf(sy_cl_Typerep_Otyperep,type,
typerep:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
quickc658316121487927005ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
quickc3360725361186068524ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > assn ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oin__range,type,
in_range: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oproper,type,
proper: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > $o ).
thf(sy_c_Assertions_OrelH,type,
relH: ( set @ nat ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o ).
thf(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
thf(sy_c_Assertions_Owand__assn,type,
wand_assn: assn > assn > assn ).
thf(sy_c_Assertions_Owand__raw,type,
wand_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
thf(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(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 @ ( A > B ) @ ( A > B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(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 ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
thf(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 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Ofstsp,type,
basic_fstsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A > $o ) ).
thf(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Osetlp,type,
basic_setlp:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > A > $o ) ).
thf(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osetrp,type,
basic_setrp:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > B > $o ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osndsp,type,
basic_sndsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B > $o ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
bit_un6697907153464112080or_num: num > num > num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
bit_un4773296044027857193um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__axioms,type,
boolea6902313364301356556axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Ocr__integer,type,
code_cr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Ocr__natural,type,
code_cr_natural: nat > code_natural > $o ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
code_rec_set_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Osub,type,
code_sub: num > num > code_integer ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit8047198070973881523_above:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit8119078960628432327_below:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
condit622319405099724424ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
condit16957441358409770ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
thf(sy_c_Countable_Onth__item,type,
nth_item:
!>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
thf(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: nat > nat > $o ).
thf(sy_c_Countable_Oto__nat,type,
to_nat:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Opart__equivp,type,
equiv_part_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Ocofinite,type,
cofinite:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
thf(sy_c_Filter_Ofrequently,type,
frequently:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
finite_comp_fun_idem:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding,type,
finite_folding:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_OTHE__default,type,
fun_THE_default:
!>[A: $tType] : ( A > ( A > $o ) > A ) ).
thf(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(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 ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
groups3894954378712506084id_sum:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_OEx1,type,
ex1:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
thf(sy_c_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( ( A > A ) > $o ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ONeg,type,
neg: num > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Ocr__int,type,
cr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Odup,type,
dup: int > int ).
thf(sy_c_Int_Oint_OAbs__int,type,
abs_int: ( set @ ( product_prod @ nat @ nat ) ) > int ).
thf(sy_c_Int_Oint_ORep__int,type,
rep_int: int > ( set @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Int_Osub,type,
sub: num > num > int ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__axioms,type,
semilattice_axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order__axioms,type,
semila6385135966242565138axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
lattic8678736583308907530nf_fin:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
lattic5652469242046573047tr_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
lattic4630905495605216202up_fin:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Lifting_Orel__pred__comp,type,
rel_pred_comp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key,type,
linord3483353639454293061rt_key:
!>[B: $tType,A: $tType] : ( ( ( B > A ) > ( list @ B ) > ( list @ B ) ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Othose,type,
those:
!>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
thf(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) ) ).
thf(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(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 ) ) ) ) ).
thf(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( ( set @ A ) > ( option @ ( set @ A ) ) ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
thf(sy_c_Misc_Olist__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_Misc_Olist__all__zip__rel,type,
list_all_zip_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
thf(sy_c_Misc_Omap__mmupd,type,
map_mmupd:
!>[B: $tType,A: $tType] : ( ( B > ( option @ A ) ) > ( set @ B ) > A > B > ( option @ A ) ) ).
thf(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list__rel,type,
merge_list_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omerge__rel,type,
merge_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort,type,
mergesort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(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 ) ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
thf(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Misc_Opartition__rev,type,
partition_rev:
!>[A: $tType] : ( ( A > $o ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Opartition__rev__rel,type,
partition_rev_rel:
!>[A: $tType] : ( ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Oquicksort__by__rel,type,
quicksort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Oquicksort__by__rel__rel,type,
quicksort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Misc_Oremove__rev,type,
remove_rev:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg__rel,type,
revg_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( option @ A ) ) ).
thf(sy_c_Misc_Oslice,type,
slice:
!>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Osu__rel__fun,type,
su_rel_fun:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_Misc_Othe__default,type,
the_default:
!>[A: $tType] : ( A > ( option @ A ) > A ) ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Misc_Ozipf,type,
zipf:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
thf(sy_c_Misc_Ozipf__rel,type,
zipf_rel:
!>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
comm_m7189776963980413722m_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mset,type,
comm_monoid_mset:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocr__multiset,type,
cr_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ofilter__mset,type,
filter_mset:
!>[A: $tType] : ( ( A > $o ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
thf(sy_c_Multiset_Oimage__mset,type,
image_mset:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
linord6283353356039996273ltiset:
!>[A: $tType] : ( ( multiset @ A ) > ( list @ A ) ) ).
thf(sy_c_Multiset_Oms__strict,type,
ms_strict: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Oms__weak,type,
ms_weak: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omulteqp__code,type,
multeqp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultiset_OAbs__multiset,type,
abs_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
thf(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultp__code,type,
multp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( C > nat ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Opw__leq,type,
pw_leq: ( multiset @ ( product_prod @ nat @ nat ) ) > ( multiset @ ( product_prod @ nat @ nat ) ) > $o ).
thf(sy_c_Multiset_Orel__mset,type,
rel_mset:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( multiset @ A ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Orepeat__mset,type,
repeat_mset:
!>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Multiset_Osize__multiset,type,
size_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred2: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > nat ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OCase,type,
old_Case:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( set @ ( old_node @ A @ B ) ) > C ) > ( ( set @ ( old_node @ A @ B ) ) > C ) > ( set @ ( old_node @ A @ B ) ) > C ) ).
thf(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OLeaf,type,
old_Leaf:
!>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OLim,type,
old_Lim:
!>[B: $tType,A: $tType] : ( ( B > ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_ONode,type,
old_Node:
!>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_ONumb,type,
old_Numb:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OPush,type,
old_Push:
!>[B: $tType] : ( ( sum_sum @ B @ nat ) > ( nat > ( sum_sum @ B @ nat ) ) > nat > ( sum_sum @ B @ nat ) ) ).
thf(sy_c_Old__Datatype_OPush__Node,type,
old_Push_Node:
!>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
thf(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 ) ) ) ).
thf(sy_c_Old__Datatype_OSplit,type,
old_Split:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > C ) > ( set @ ( old_node @ A @ B ) ) > C ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
old_Abs_Node:
!>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
thf(sy_c_Old__Datatype_Onode_ORep__Node,type,
old_Rep_Node:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_Old__Datatype_Ousum,type,
old_usum:
!>[A: $tType,B: $tType] : ( ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder_OGreatest,type,
greatest:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
thf(sy_c_Partial__Function_Omk__less,type,
partial_mk_less:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Predicate_OSeq,type,
seq2:
!>[A: $tType] : ( ( product_unit > ( seq @ A ) ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oadjunct,type,
adjunct:
!>[A: $tType] : ( ( pred @ A ) > ( seq @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oapply,type,
apply:
!>[A: $tType,B: $tType] : ( ( A > ( pred @ B ) ) > ( seq @ A ) > ( seq @ B ) ) ).
thf(sy_c_Predicate_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Ocontained,type,
contained:
!>[A: $tType] : ( ( seq @ A ) > ( pred @ A ) > $o ) ).
thf(sy_c_Predicate_Oif__pred,type,
if_pred: $o > ( pred @ product_unit ) ).
thf(sy_c_Predicate_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( pred @ A ) > $o ) ).
thf(sy_c_Predicate_Oiterate__upto,type,
iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oiterate__upto__rel,type,
iterate_upto_rel:
!>[A: $tType] : ( ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > $o ) ).
thf(sy_c_Predicate_Omap,type,
map2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( pred @ A ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Onot__pred,type,
not_pred: ( pred @ product_unit ) > ( pred @ product_unit ) ).
thf(sy_c_Predicate_Onull,type,
null:
!>[A: $tType] : ( ( seq @ A ) > $o ) ).
thf(sy_c_Predicate_Opred_OPred,type,
pred3:
!>[A: $tType] : ( ( A > $o ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Opred_Oeval,type,
eval:
!>[A: $tType] : ( ( pred @ A ) > A > $o ) ).
thf(sy_c_Predicate_Opred__of__seq,type,
pred_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Opred__of__set,type,
pred_of_set:
!>[A: $tType] : ( ( set @ A ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : ( seq @ A ) ).
thf(sy_c_Predicate_Oseq_OInsert,type,
insert:
!>[A: $tType] : ( A > ( pred @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oseq_OJoin,type,
join:
!>[A: $tType] : ( ( pred @ A ) > ( seq @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oseq_Ocase__seq,type,
case_seq:
!>[B: $tType,A: $tType] : ( B > ( A > ( pred @ A ) > B ) > ( ( pred @ A ) > ( seq @ A ) > B ) > ( seq @ A ) > B ) ).
thf(sy_c_Predicate_Oset__of__pred,type,
set_of_pred:
!>[A: $tType] : ( ( pred @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Oset__of__seq,type,
set_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Osingle,type,
single:
!>[A: $tType] : ( A > ( pred @ A ) ) ).
thf(sy_c_Predicate_Osingleton,type,
singleton:
!>[A: $tType] : ( ( product_unit > A ) > ( pred @ A ) > A ) ).
thf(sy_c_Predicate_Othe,type,
the3:
!>[A: $tType] : ( ( pred @ A ) > A ) ).
thf(sy_c_Predicate_Othe__only,type,
the_only:
!>[A: $tType] : ( ( product_unit > A ) > ( seq @ A ) > A ) ).
thf(sy_c_Predicate__Compile_Ocontains,type,
predicate_contains:
!>[A: $tType] : ( ( set @ A ) > A > $o ) ).
thf(sy_c_Predicate__Compile_Ocontains__pred,type,
predic7144156976422707464s_pred:
!>[A: $tType] : ( ( set @ A ) > A > ( pred @ product_unit ) ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Random_Oinc__shift,type,
inc_shift: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(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 ) ) ).
thf(sy_c_Random__Pred_ORandom,type,
random_Random:
!>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( A > ( 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 ) ) ) ).
thf(sy_c_Random__Pred_Oempty,type,
random_empty:
!>[A: $tType] : ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oiterate__upto,type,
random_iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Onot__randompred,type,
random6974930770145893639ompred: ( ( 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 ) ) ).
thf(sy_c_Random__Pred_Osingle,type,
random_single:
!>[A: $tType] : ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( ( 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 ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ocr__rat,type,
cr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Orat_OAbs__rat,type,
abs_rat: ( set @ ( product_prod @ int @ int ) ) > rat ).
thf(sy_c_Rat_Orat_ORep__rat,type,
rep_rat: rat > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(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 ) ) ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
set_atLeastLessThan:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThan,type,
set_greaterThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
set_gr3752724095348155675AtMost:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr287244882034783167ssThan:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OlessThan,type,
set_lessThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_OCode_Oabort,type,
abort:
!>[A: $tType] : ( literal > ( product_unit > A ) > A ) ).
thf(sy_c_String_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Oasciis__of__literal,type,
asciis_of_literal: literal > ( list @ code_integer ) ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Odigit7,type,
digit7: char > $o ).
thf(sy_c_String_Ochar__of__integer,type,
char_of_integer: code_integer > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ocr__literal,type,
cr_literal: ( list @ char ) > literal > $o ).
thf(sy_c_String_Oliteral_OAbs__literal,type,
abs_literal: ( list @ char ) > literal ).
thf(sy_c_String_Oliteral_Oexplode,type,
explode: literal > ( list @ char ) ).
thf(sy_c_String_Opcr__literal,type,
pcr_literal: ( list @ char ) > literal > $o ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
sum_rec_set_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T > $o ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Sum__Type_Osum_Oprojl,type,
sum_projl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > A ) ).
thf(sy_c_Sum__Type_Osum_Oprojr,type,
sum_projr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > B ) ).
thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__total,type,
left_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Otransfer__bforall,type,
transfer_bforall:
!>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(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 ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
thf(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
pred_s596693808085603175closed:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
pred_s7749564232668923593losedp:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_x,type,
x: assn ).
thf(sy_v_y,type,
y: assn ).
% Relevant facts (7782)
thf(fact_0_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_1_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_2_Abs__assn__eqI_I2_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_3_Abs__assn__eqI_I1_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_4_wand__assn__def,axiom,
( wand_assn
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% wand_assn_def
thf(fact_5_Abs__assn__inverse,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( rep_assn @ ( abs_assn @ Y ) )
= Y ) ) ).
% Abs_assn_inverse
thf(fact_6_inf__assn__def,axiom,
( ( inf_inf @ assn )
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H2 )
& ( rep_assn @ Q @ H2 ) ) ) ) ) ).
% inf_assn_def
thf(fact_7_sup__assn__def,axiom,
( ( sup_sup @ assn )
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H2 )
| ( rep_assn @ Q @ H2 ) ) ) ) ) ).
% sup_assn_def
thf(fact_8_bot__assn__def,axiom,
( ( bot_bot @ assn )
= ( abs_assn
@ ^ [Uu: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ) ).
% bot_assn_def
thf(fact_9_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( X
= ( abs_assn @ Y2 ) )
=> ~ ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y2 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ) ) ).
% Abs_assn_cases
thf(fact_10_Abs__assn__induct,axiom,
! [P: assn > $o,X: assn] :
( ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y2 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( P @ ( abs_assn @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_assn_induct
thf(fact_11_Abs__assn__inject,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ X @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( ( abs_assn @ X )
= ( abs_assn @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_assn_inject
thf(fact_12_Rep__assn,axiom,
! [X: assn] : ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( rep_assn @ X ) @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ) ).
% Rep_assn
thf(fact_13_Rep__assn__cases,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ~ ! [X2: assn] :
( Y
!= ( rep_assn @ X2 ) ) ) ).
% Rep_assn_cases
thf(fact_14_Rep__assn__induct,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,P: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ! [X2: assn] : ( P @ ( rep_assn @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_assn_induct
thf(fact_15_times__assn__def,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% times_assn_def
thf(fact_16_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ).
% bool_assn_proper(2)
thf(fact_17_bool__assn__proper_I3_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H2 )
| ( Q2 @ H2 ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_18_bool__assn__proper_I4_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H2 )
& ( Q2 @ H2 ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_19_wand__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] : ( proper @ ( wand_raw @ P @ Q2 ) ) ).
% wand_proper
thf(fact_20_times__assn__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper @ ( times_assn_raw @ P @ Q2 ) ) ) ) ).
% times_assn_proper
thf(fact_21_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q: assn] : ( times_times @ assn @ Q @ P2 ) ) ) ).
% assn_times_comm
thf(fact_22_assn__times__assoc,axiom,
! [P: assn,Q2: assn,R: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q2 ) @ R )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q2 @ R ) ) ) ).
% assn_times_assoc
thf(fact_23_mod__starE,axiom,
! [A3: assn,B2: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H3 )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A3 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B2 @ H_2 ) ) ) ).
% mod_starE
thf(fact_24_mod__starD,axiom,
! [A4: assn,B3: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A4 @ B3 ) @ H3 )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A4 @ H1 )
& ( rep_assn @ B3 @ H22 ) ) ) ).
% mod_starD
thf(fact_25_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_26_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_27_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_28_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_29_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_30_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_31_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( sup_sup @ A @ A3 @ B2 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_32_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_33_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B2 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_34_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_35_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_36_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_37_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X3: A] : ( inf_inf @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% inf_apply
thf(fact_38_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_39_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_40_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_41_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_42_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_43_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_44_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G2 @ X2 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_50_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_51_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_52_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_53_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_54_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X3: A] : ( inf_inf @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% inf_fun_def
thf(fact_55_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_56_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C2 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_57_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_commute
thf(fact_58_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B4: A] : ( inf_inf @ A @ B4 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_59_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_60_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_61_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_62_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_63_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_64_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_65_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_66_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_67_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_68_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_69_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B4: A] : ( sup_sup @ A @ B4 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_70_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_71_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_72_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_73_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_74_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_75_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_76_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_77_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_inf_distrib1
thf(fact_78_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_79_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_distrib1
thf(fact_80_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z3: A] :
( ( sup_sup @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ ( sup_sup @ A @ X2 @ Z3 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp2
thf(fact_81_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z3: A] :
( ( inf_inf @ A @ X2 @ ( sup_sup @ A @ Y2 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ ( inf_inf @ A @ X2 @ Z3 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp1
thf(fact_82_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_83_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_84_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_85_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_86_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_87_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_88_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_89_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_90_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_91_type__definition__assn,axiom,
type_definition @ assn @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ rep_assn @ abs_assn @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ).
% type_definition_assn
thf(fact_92_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A3 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_93_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A4 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_94_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_95_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A4 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_96_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A3 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_97_type__definition_ORep,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( member @ A @ ( Rep @ X ) @ A4 ) ) ).
% type_definition.Rep
thf(fact_98_type__definition_Ointro,axiom,
! [B: $tType,A: $tType,Rep: B > A,A4: set @ A,Abs: A > B] :
( ! [X2: B] : ( member @ A @ ( Rep @ X2 ) @ A4 )
=> ( ! [X2: B] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_definition @ B @ A @ Rep @ Abs @ A4 ) ) ) ) ).
% type_definition.intro
thf(fact_99_type__definition_OAbs__cases,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ~ ! [Y2: A] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member @ A @ Y2 @ A4 ) ) ) ).
% type_definition.Abs_cases
thf(fact_100_type__definition_ORep__cases,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A,Y: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ~ ! [X2: B] :
( Y
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_101_type__definition_OAbs__induct,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A,P: B > $o,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_102_type__definition_OAbs__inject,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,X: A,Y: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_103_type__definition_ORep__induct,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,Y: A,P: A > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ! [X2: B] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_104_type__definition_ORep__inject,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A,X: B,Y: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_105_type__definition_OAbs__inverse,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,Y: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_106_type__definition_ORep__inverse,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_107_type__definition__def,axiom,
! [A: $tType,B: $tType] :
( ( type_definition @ B @ A )
= ( ^ [Rep2: B > A,Abs2: A > B,A6: set @ A] :
( ! [X3: B] : ( member @ A @ ( Rep2 @ X3 ) @ A6 )
& ! [X3: B] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_108_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.right_assoc
thf(fact_109_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% mult.right_commute
thf(fact_110_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_111_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ B4 @ A5 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_112_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_113_mult_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ B2 @ A3 ) ) ) ) ).
% mult.safe_commute
thf(fact_114_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
thf(fact_115_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_116_one__assn__def,axiom,
( ( one_one @ assn )
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_117_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A3 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_118_mod__star__conv,axiom,
! [A4: assn,B3: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A4 @ B3 ) @ H3 )
= ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As2: set @ nat] :
( ( H3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
& ( rep_assn @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
& ( rep_assn @ B3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As2 ) ) ) ) ) ).
% mod_star_conv
thf(fact_119_star__assnI,axiom,
! [P: assn,H3: heap_ext @ product_unit,As: set @ nat,Q2: assn,As3: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As @ As3 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As @ As3 ) ) ) ) ) ) ).
% star_assnI
thf(fact_120_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
thf(fact_121_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y ) )
= ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_122_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_123_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_124_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_125_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_126_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_127_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_128_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_129_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_130_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_131_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_132_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_133_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_134_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( inf_inf @ A @ A3 @ B2 )
= ( top_top @ A ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_135_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
thf(fact_136_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A3 @ B2 ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_137_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
= A3 ) ) ).
% inf_top.right_neutral
thf(fact_138_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_left
thf(fact_139_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_right
thf(fact_140_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_141_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_142_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_143_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_144_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_145_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_146_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_147_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_148_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_149_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_150_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_151_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_152_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_153_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y ) )
= ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_154_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_155_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.comm_monoid_axioms
thf(fact_156_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_157_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semilattice_neutr @ A @ F2 @ Z2 )
=> ( comm_monoid @ A @ F2 @ Z2 ) ) ).
% semilattice_neutr.axioms(2)
thf(fact_158_monoid_Oleft__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ Z2 @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_159_monoid_Oright__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ A3 @ Z2 )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_160_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ).
% one_assn_raw.cases
thf(fact_161_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ A3 @ Z2 )
= A3 ) ) ).
% comm_monoid.comm_neutral
thf(fact_162_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_163_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_164_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_165_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_166_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_167_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
thf(fact_168_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.monoid_axioms
thf(fact_169_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_170_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_171_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
thf(fact_172_syntax__fo__nomatch__def,axiom,
! [B: $tType,A: $tType] :
( ( syntax7388354845996824322omatch @ A @ B )
= ( ^ [Pat: A,Obj: B] : $true ) ) ).
% syntax_fo_nomatch_def
thf(fact_173_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% boolean_algebra.conj_one_right
thf(fact_174_mod__h__bot__indep,axiom,
! [P: assn,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_175_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_176_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( As4
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_177_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( As4
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_178_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( Y
= ( As4
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_179_one__assn__raw_Osimps,axiom,
! [H3: heap_ext @ product_unit,As: set @ nat] :
( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
= ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% one_assn_raw.simps
thf(fact_180_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As: set @ nat] :
( ( times_assn_raw @ P @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
= ( ? [As1: set @ nat,As2: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As1 ) )
& ( Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_181_times__assn__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( Y
= ( ~ ? [As1: set @ nat,As2: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_182_times__assn__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ~ ? [As12: set @ nat,As22: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As12 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_183_times__assn__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ? [As13: set @ nat,As23: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As13 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_184_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_185_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_186_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_187_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z2 )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1
thf(fact_188_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1_right
thf(fact_189_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_190_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_191_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( inf_inf @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_192_Un__Int__eq_I4_J,axiom,
! [A: $tType,T2: set @ A,S: set @ A] :
( ( inf_inf @ ( set @ A ) @ T2 @ ( sup_sup @ ( set @ A ) @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_193_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_194_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_195_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( sup_sup @ ( set @ A ) @ S @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_196_Int__Un__eq_I4_J,axiom,
! [A: $tType,T2: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T2 @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_197_Un__empty,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_198_Compl__disjoint2,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_199_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_200_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_201_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_202_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_203_IntI,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% IntI
thf(fact_204_Int__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C2 @ A4 )
& ( member @ A @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_205_Int__UNIV,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( top_top @ ( set @ A ) ) )
= ( ( A4
= ( top_top @ ( set @ A ) ) )
& ( B3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_206_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A4 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnCI
thf(fact_207_Un__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C2 @ A4 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_208_ComplI,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_209_Compl__iff,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C2 @ A4 ) ) ) ).
% Compl_iff
thf(fact_210_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( A4 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_211_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S2: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S2: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_212_Compl__disjoint,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_213_ComplD,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C2 @ A4 ) ) ).
% ComplD
thf(fact_214_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_215_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_216_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_217_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_218_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_219_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_220_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_221_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_222_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_223_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_224_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_225_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_226_IntE,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ~ ( member @ A @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_227_IntD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% IntD1
thf(fact_228_IntD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C2 @ B3 ) ) ).
% IntD2
thf(fact_229_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Int_assoc
thf(fact_230_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_231_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Int_commute
thf(fact_232_Int__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_233_Int__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= A4 ) ).
% Int_UNIV_right
thf(fact_234_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Int_left_absorb
thf(fact_235_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_236_UnE,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_237_UnI1,axiom,
! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnI1
thf(fact_238_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnI2
thf(fact_239_bex__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B3 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_240_ball__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_241_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_assoc
thf(fact_242_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_243_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Un_commute
thf(fact_244_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_245_Un__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_246_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_left_absorb
thf(fact_247_Compl__partition,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_248_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_249_Compl__partition2,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_250_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% Set.empty_def
thf(fact_251_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_252_Int__Collect,axiom,
! [A: $tType,X: A,A4: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A4 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_253_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_254_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_255_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A6 )
| ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_256_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_257_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_258_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_259_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_260_Int__emptyI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ~ ( member @ A @ X2 @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_261_disjoint__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ~ ( member @ A @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_262_Int__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_263_Int__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_264_disjoint__iff__not__equal,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B3 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_265_Un__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_266_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_267_Compl__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% Compl_Un
thf(fact_268_Compl__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% Compl_Int
thf(fact_269_Un__Int__crazy,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) @ ( inf_inf @ ( set @ A ) @ C3 @ A4 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) @ ( sup_sup @ ( set @ A ) @ C3 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_270_Int__Un__distrib,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_271_Un__Int__distrib,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_272_Int__Un__distrib2,axiom,
! [A: $tType,B3: set @ A,C3: set @ A,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C3 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_273_Un__Int__distrib2,axiom,
! [A: $tType,B3: set @ A,C3: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) @ A4 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ A4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_274_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_275_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_276_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_277_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_278_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_279_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B6 ) )
= ( ( A3 = A7 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_280_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_281_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X3: A] : X3 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_282_disjointI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ~ ( member @ A @ X2 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_283_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_284_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_285_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_286_inf1I,axiom,
! [A: $tType,A4: A > $o,X: A,B3: A > $o] :
( ( A4 @ X )
=> ( ( B3 @ X )
=> ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X ) ) ) ).
% inf1I
thf(fact_287_sup1CI,axiom,
! [A: $tType,B3: A > $o,X: A,A4: A > $o] :
( ( ~ ( B3 @ X )
=> ( A4 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X ) ) ).
% sup1CI
thf(fact_288_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_289_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_290_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_291_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_292_sup1I2,axiom,
! [A: $tType,B3: A > $o,X: A,A4: A > $o] :
( ( B3 @ X )
=> ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X ) ) ).
% sup1I2
thf(fact_293_sup1I1,axiom,
! [A: $tType,A4: A > $o,X: A,B3: A > $o] :
( ( A4 @ X )
=> ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X ) ) ).
% sup1I1
thf(fact_294_inf1D2,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X )
=> ( B3 @ X ) ) ).
% inf1D2
thf(fact_295_inf1D1,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X )
=> ( A4 @ X ) ) ).
% inf1D1
thf(fact_296_sup1E,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X )
=> ( ~ ( A4 @ X )
=> ( B3 @ X ) ) ) ).
% sup1E
thf(fact_297_inf1E,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X )
=> ~ ( ( A4 @ X )
=> ~ ( B3 @ X ) ) ) ).
% inf1E
thf(fact_298_times__assn__raw_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [P3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As4: set @ nat] :
( X
!= ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) ) ).
% times_assn_raw.cases
thf(fact_299_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F3: A > B,A8: A,B7: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F3 @ ( product_Pair @ A @ A @ A8 @ B7 ) ) ) ).
% pairself.cases
thf(fact_300_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A8: A,B7: B] :
( Y
!= ( product_Pair @ A @ B @ A8 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_301_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( P @ A3 @ B2 ) )
=> ? [A8: A,B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ S )
& ( P @ A8 @ B7 ) ) ) ) ).
% bex2I
thf(fact_302_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X2: A,Y2: B] :
( P4
= ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_303_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A8: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_304_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B6 ) )
=> ~ ( ( A3 = A7 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_305_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A8: A,B7: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% prod_cases3
thf(fact_306_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A8: A,B7: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B7 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_307_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 ) ) )] :
~ ! [A8: A,B7: B,C4: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_308_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A8: A,B7: B,C4: C,D2: D,E2: E,F3: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_309_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A8: A,B7: B,C4: C,D2: D,E2: E,F3: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_310_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A8: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_311_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A8: A,B7: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B7 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_312_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( 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 ) ) )] :
( ! [A8: A,B7: B,C4: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_313_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A8: A,B7: B,C4: C,D2: D,E2: E,F3: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_314_prod__induct7,axiom,
! [G3: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A8: A,B7: B,C4: C,D2: D,E2: E,F3: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_315_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
thf(fact_316_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
~ ( member @ A @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_317_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_318_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_319_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_320_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_321_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_322_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_323_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_324_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F5: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ? [X4: B] : ( F5 @ X4 ) )
= ( ? [B4: A] : ( F5 @ ( Rep @ B4 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_325_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S3: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X2: B] :
( S3
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_326_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_327_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
thf(fact_328_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A3: B,B2: C] :
( ( uncurry @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F2 @ A3 @ B2 ) ) ).
% uncurry_apply
thf(fact_329_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_330_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_331_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_332_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ One )
= Zero ) ) ).
% abstract_boolean_algebra.compl_one
thf(fact_333_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ Zero )
= One ) ) ).
% abstract_boolean_algebra.compl_zero
thf(fact_334_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ X @ Y )
= Zero )
=> ( ( ( Disj @ X @ Y )
= One )
=> ( ( Compl @ X )
= Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
thf(fact_335_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Compl @ X ) )
= X ) ) ).
% abstract_boolean_algebra.double_compl
thf(fact_336_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ One @ X )
= One ) ) ).
% abstract_boolean_algebra.disj_one_left
thf(fact_337_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ One )
= X ) ) ).
% abstract_boolean_algebra.conj_one_right
thf(fact_338_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ Zero @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
thf(fact_339_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Conj @ X @ Y ) )
= ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
thf(fact_340_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Disj @ X @ Y ) )
= ( Conj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
thf(fact_341_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ One )
= One ) ) ).
% abstract_boolean_algebra.disj_one_right
thf(fact_342_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ Zero )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
thf(fact_343_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ Zero )
= X ) ) ).
% abstract_boolean_algebra.disj_zero_right
thf(fact_344_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ ( Compl @ X ) @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
thf(fact_345_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ ( Compl @ X ) @ X )
= One ) ) ).
% abstract_boolean_algebra.disj_cancel_left
thf(fact_346_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,A3: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ A3 @ X )
= Zero )
=> ( ( ( Disj @ A3 @ X )
= One )
=> ( ( ( Conj @ A3 @ Y )
= Zero )
=> ( ( ( Disj @ A3 @ Y )
= One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
thf(fact_347_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ ( Compl @ X ) )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
thf(fact_348_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A,Z2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ ( Disj @ Y @ Z2 ) )
= ( Disj @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
thf(fact_349_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ ( Compl @ X ) )
= One ) ) ).
% abstract_boolean_algebra.disj_cancel_right
thf(fact_350_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A,Z2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ ( Conj @ Y @ Z2 ) )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
thf(fact_351_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Compl @ X )
= ( Compl @ Y ) )
= ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
thf(fact_352_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Y: A,Z2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ ( Disj @ Y @ Z2 ) @ X )
= ( Disj @ ( Conj @ Y @ X ) @ ( Conj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
thf(fact_353_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Y: A,Z2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ ( Conj @ Y @ Z2 ) @ X )
= ( Conj @ ( Disj @ Y @ X ) @ ( Disj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
thf(fact_354_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_355_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_356_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_357_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_358_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_359_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_360_times__assn__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) )
=> ? [As13: set @ nat,As23: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As13 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_361_times__assn__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) )
=> ~ ? [As12: set @ nat,As22: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As12 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_362_times__assn__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( Y
= ( ? [As1: set @ nat,As2: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_363_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty2 @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_364_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C2 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_365_wand__assnI,axiom,
! [H3: heap_ext @ product_unit,As: set @ nat,Q2: assn,R: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( ! [H5: heap_ext @ product_unit,As5: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As @ H3 @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
=> ( ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As5 ) )
=> ( rep_assn @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As @ As5 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q2 @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ).
% wand_assnI
thf(fact_366_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A4: set @ B,D3: B,B3: set @ A] :
( ( ( member @ B @ C2 @ A4 )
=> ( ( ( member @ B @ D3 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ B3 ) @ C2 @ D3 )
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ B3 ) @ C2 @ D3 )
@ A4 )
= B3 ) ) ) )
& ( ~ ( member @ B @ C2 @ A4 )
=> ( ( ( member @ B @ D3 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ B3 ) @ C2 @ D3 )
@ A4 )
= ( uminus_uminus @ ( set @ A ) @ B3 ) ) )
& ( ~ ( member @ B @ D3 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ B3 ) @ C2 @ D3 )
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_367_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
! [A: $tType] : ( type_definition @ A @ A @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ A ) @ ( top_top @ ( set @ A ) ) ) ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_368_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)
thf(fact_369_wand__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ! [H5: heap_ext @ product_unit,As5: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As5 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As4 @ As5 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_370_wand__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ! [H6: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As4 @ As6 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_371_sup2CI,axiom,
! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y: B,A4: A > B > $o] :
( ( ~ ( B3 @ X @ Y )
=> ( A4 @ X @ Y ) )
=> ( sup_sup @ ( A > B > $o ) @ A4 @ B3 @ X @ Y ) ) ).
% sup2CI
thf(fact_372_inf2I,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: B,B3: A > B > $o] :
( ( A4 @ X @ Y )
=> ( ( B3 @ X @ Y )
=> ( inf_inf @ ( A > B > $o ) @ A4 @ B3 @ X @ Y ) ) ) ).
% inf2I
thf(fact_373_vimage__ident,axiom,
! [A: $tType,Y4: set @ A] :
( ( vimage @ A @ A
@ ^ [X3: A] : X3
@ Y4 )
= Y4 ) ).
% vimage_ident
thf(fact_374_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,P: B > $o] :
( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y3: A] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_375_vimage__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_376_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_377_vimage__Int,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_Int
thf(fact_378_vimage__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_Un
thf(fact_379_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_380_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_381_relH__dist__union,axiom,
! [As: set @ nat,As3: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ ( sup_sup @ ( set @ nat ) @ As @ As3 ) @ H3 @ H4 )
= ( ( relH @ As @ H3 @ H4 )
& ( relH @ As3 @ H3 @ H4 ) ) ) ).
% relH_dist_union
thf(fact_382_bool__assn__proper_I5_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H2 )
& ~ ( P @ H2 ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_383_in__range__empty,axiom,
! [H3: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% in_range_empty
thf(fact_384_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A4: set @ B] :
( ( ( member @ B @ C2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : C2
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : C2
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_385_in__range__dist__union,axiom,
! [H3: heap_ext @ product_unit,As: set @ nat,As3: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As @ As3 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ).
% in_range_dist_union
thf(fact_386_relH__sym,axiom,
! [As: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As @ H3 @ H4 )
=> ( relH @ As @ H4 @ H3 ) ) ).
% relH_sym
thf(fact_387_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
thf(fact_388_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_389_inf2E,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B3: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A4 @ B3 @ X @ Y )
=> ~ ( ( A4 @ X @ Y )
=> ~ ( B3 @ X @ Y ) ) ) ).
% inf2E
thf(fact_390_sup2E,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B3: A > B > $o,X: A,Y: B] :
( ( sup_sup @ ( A > B > $o ) @ A4 @ B3 @ X @ Y )
=> ( ~ ( A4 @ X @ Y )
=> ( B3 @ X @ Y ) ) ) ).
% sup2E
thf(fact_391_inf2D1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B3: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A4 @ B3 @ X @ Y )
=> ( A4 @ X @ Y ) ) ).
% inf2D1
thf(fact_392_inf2D2,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B3: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A4 @ B3 @ X @ Y )
=> ( B3 @ X @ Y ) ) ).
% inf2D2
thf(fact_393_sup2I1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: B,B3: A > B > $o] :
( ( A4 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A4 @ B3 @ X @ Y ) ) ).
% sup2I1
thf(fact_394_sup2I2,axiom,
! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y: B,A4: A > B > $o] :
( ( B3 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A4 @ B3 @ X @ Y ) ) ).
% sup2I2
thf(fact_395_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F: A > B,B5: set @ B] :
( collect @ A
@ ^ [X3: A] : ( member @ B @ ( F @ X3 ) @ B5 ) ) ) ) ).
% vimage_def
thf(fact_396_relH__in__rangeI_I2_J,axiom,
! [As: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As @ H3 @ H4 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ).
% relH_in_rangeI(2)
thf(fact_397_relH__in__rangeI_I1_J,axiom,
! [As: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As @ H3 @ H4 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ).
% relH_in_rangeI(1)
thf(fact_398_relH__refl,axiom,
! [H3: heap_ext @ product_unit,As: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( relH @ As @ H3 @ H3 ) ) ).
% relH_refl
thf(fact_399_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B,G2: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G2 @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_400_vimage__Compl,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( vimage @ A @ B @ F2 @ ( uminus_uminus @ ( set @ B ) @ A4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) ) ) ).
% vimage_Compl
thf(fact_401_proper__iff,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( relH @ As @ H3 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
= ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ) ).
% proper_iff
thf(fact_402_proper__def,axiom,
( proper
= ( ^ [P2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
! [H2: heap_ext @ product_unit,H7: heap_ext @ product_unit,As7: set @ nat] :
( ( ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As7 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As7 ) ) )
& ( ( ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As7 ) )
& ( relH @ As7 @ H2 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As7 ) ) )
=> ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As7 ) ) ) ) ) ) ).
% proper_def
thf(fact_403_properD2,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As: set @ nat,H4: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( ( relH @ As @ H3 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ) ).
% properD2
thf(fact_404_properI,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [As4: set @ nat,H: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) )
=> ( ! [As4: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( relH @ As4 @ H @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As4 ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As4 ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_405_models__in__range,axiom,
! [P: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H3 )
=> ( in_range @ H3 ) ) ).
% models_in_range
thf(fact_406_top__assn__def,axiom,
( ( top_top @ assn )
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_407_properD1,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As: set @ nat] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ).
% properD1
thf(fact_408_mod__relH,axiom,
! [As: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,P: assn] :
( ( relH @ As @ H3 @ H4 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ).
% mod_relH
thf(fact_409_uminus__assn__def,axiom,
( ( uminus_uminus @ assn )
= ( ^ [P2: assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H2 )
& ~ ( rep_assn @ P2 @ H2 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_410_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As: set @ nat] :
( ( wand_raw @ P @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H3 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_411_wand__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As4 @ As8 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_412_wand__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As4 @ As8 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_413_wand__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ! [H6: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As4 @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_414_wand__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ! [H5: heap_ext @ product_unit,As5: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As5 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As4 @ As5 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_415_reflclp__idemp,axiom,
! [A: $tType,P: A > A > $o] :
( ( sup_sup @ ( A > A > $o )
@ ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% reflclp_idemp
thf(fact_416_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_417_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_418_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_419_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_dec_simps(2)
thf(fact_420_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ Xa ) )
=> ~ ! [A8: A,B7: A] :
( ( Xa
= ( product_Pair @ A @ A @ A8 @ B7 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A8 ) @ ( X @ B7 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ ( product_Pair @ A @ A @ A8 @ B7 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_421_rel__restrict__compl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A4 ) @ ( rel_restrict @ A @ R @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_422_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_423_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_424_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_425_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_426_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_427_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_428_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V: num,W2: num,Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Z2 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_429_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_430_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_431_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_432_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_433_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_434_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_435_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_436_rel__restrict__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R @ ( bot_bot @ ( set @ A ) ) )
= R ) ).
% rel_restrict_empty
thf(fact_437_Id__onI,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A4 ) ) ) ).
% Id_onI
thf(fact_438_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_439_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_440_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_441_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_442_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_443_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_444_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_445_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)
thf(fact_446_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_447_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_448_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G2: C] :
( ( F2
= ( ^ [X3: A > B] : G2 ) )
=> ( ( F2
@ ^ [X3: A] : ( zero_zero @ B ) )
= G2 ) ) ) ).
% fun_cong_unused_0
thf(fact_449_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_450_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_451_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_452_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_453_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_454_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_455_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_456_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_457_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E3 @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E3 ) ) ).
% rel_restrict_lift
thf(fact_458_rel__restrictI,axiom,
! [A: $tType,X: A,R: set @ A,Y: A,E3: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ R )
=> ( ~ ( member @ A @ Y @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E3 @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_459_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R ) )
=> ~ ( member @ A @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_460_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R ) )
=> ~ ( member @ A @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_461_rel__restrict__union,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( rel_restrict @ A @ R @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R @ A4 ) @ B3 ) ) ).
% rel_restrict_union
thf(fact_462_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A4 ) )
= ( ( X = Y )
& ( member @ A @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_463_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A4: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_464_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A4 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( C2
!= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_465_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_466_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_467_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_468_zero__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% zero_neq_neg_one
thf(fact_469_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,B2: A] :
( ( pairself @ A @ B @ F2 @ ( product_Pair @ A @ A @ A3 @ B2 ) )
= ( product_Pair @ B @ B @ ( F2 @ A3 ) @ ( F2 @ B2 ) ) ) ).
% pairself.simps
thf(fact_470_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa )
= Y )
=> ~ ! [A8: A,B7: A] :
( ( Xa
= ( product_Pair @ A @ A @ A8 @ B7 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A8 ) @ ( X @ B7 ) ) ) ) ) ).
% pairself.elims
thf(fact_471_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z5: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R2 ) )
=> ( Y3 = Z5 ) )
& ! [X3: A,Y3: A,Z5: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R2 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R2 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_472_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_473_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W2: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_474_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_475_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W2: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_476_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_477_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_478_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_479_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_480_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_481_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: A,R3: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R3 ) )
= ( R3
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_482_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A5: assn,B4: assn] : ( inf_inf @ assn @ A5 @ ( uminus_uminus @ assn @ B4 ) ) ) ) ).
% minus_assn_def
thf(fact_483_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A6 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% minus_apply
thf(fact_484_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_485_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_486_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_487_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_488_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_489_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_490_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_491_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_492_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_493_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_494_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
= ( uminus_uminus @ B @ B2 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_495_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_496_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_497_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_498_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_499_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% iszero_neg_numeral
thf(fact_500_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_501_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_502_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(12)
thf(fact_503_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A6 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% fun_diff_def
thf(fact_504_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_505_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( A3 = B2 )
= ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_506_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_507_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_508_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_509_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_510_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_511_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_512_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_513_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% diff_eq
thf(fact_514_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_515_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_516_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_517_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_518_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_519_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_520_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_521_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X ) ) ) ).
% divide_minus1
thf(fact_522_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_523_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
thf(fact_524_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
thf(fact_525_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_526_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_527_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
thf(fact_528_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
thf(fact_529_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_530_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_531_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_532_Un__Diff__cancel,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_533_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ B3 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_534_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_535_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 @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_536_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 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_537_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_538_Diff__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_539_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_540_Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_541_Diff__Compl,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_Compl
thf(fact_542_inter__compl__diff__conv,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% inter_compl_diff_conv
thf(fact_543_Compl__Diff__eq,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ B3 ) ) ).
% Compl_Diff_eq
thf(fact_544_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_545_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_546_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_547_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_548_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_549_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_550_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_551_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_552_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
thf(fact_553_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
thf(fact_554_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Int_Diff
thf(fact_555_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_556_Diff__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_557_Diff__Int__distrib,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C3 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_558_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_559_Un__Diff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_Diff
thf(fact_560_set__diff__diff__left,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% set_diff_diff_left
thf(fact_561_Diff__triv,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
= A4 ) ) ).
% Diff_triv
thf(fact_562_Int__Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_563_disjoint__alt__simp1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
= A4 )
= ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_564_disjoint__alt__simp2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
!= A4 )
= ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_565_Diff__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Diff_Un
thf(fact_566_Diff__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Diff_Int
thf(fact_567_Int__Diff__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_568_Un__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_569_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_570_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 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_571_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 @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_572_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z2: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_573_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_574_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_575_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_576_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_577_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 ) )
= ( ( times_times @ A @ A3 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_578_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
= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_579_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_580_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_581_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_582_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_583_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 ) )
= ( ( times_times @ A @ X @ Z2 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_584_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
thf(fact_585_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_586_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_587_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_588_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_589_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_590_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_591_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_592_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_593_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_594_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_595_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_596_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_597_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_598_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_599_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
thf(fact_600_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q2: A > $o] :
( ! [X2: A,K2: A] :
( ( P @ X2 )
= ( P @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X2: A,K2: A] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X5: A,K3: A] :
( ( ( P @ X5 )
& ( Q2 @ X5 ) )
= ( ( P @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K3 @ D4 ) ) )
& ( Q2 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K3 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_601_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q2: A > $o] :
( ! [X2: A,K2: A] :
( ( P @ X2 )
= ( P @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X2: A,K2: A] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X5: A,K3: A] :
( ( ( P @ X5 )
| ( Q2 @ X5 ) )
= ( ( P @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K3 @ D4 ) ) )
| ( Q2 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K3 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_602_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_603_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_604_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_605_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_606_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_607_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_608_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_609_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_610_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_611_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_612_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_613_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_614_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_615_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_616_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_617_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_618_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_619_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_620_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_621_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_622_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_623_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_624_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_625_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_626_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_627_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_628_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_629_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_630_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_631_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_632_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_633_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_634_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_635_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_636_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_637_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_638_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_639_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_640_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_641_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_642_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_643_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_644_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_645_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_646_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_647_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_648_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_649_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_650_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_651_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_652_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_653_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_654_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_655_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_656_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_657_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_658_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_659_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_660_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A6 )
& ~ ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_661_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_662_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_663_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_664_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_665_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_666_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_667_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_668_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_669_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_670_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_671_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_672_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_673_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_674_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_675_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_676_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_677_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_678_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_679_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_680_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_681_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_682_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_683_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( A3
= ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_684_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_685_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_686_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_687_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_688_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_689_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_690_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_691_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( A3
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_692_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_693_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_694_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ) ).
% order_less_le
thf(fact_695_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ) ).
% order_le_less
thf(fact_696_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_697_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_698_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_699_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_700_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_701_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_702_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_703_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_704_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_705_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% antisym
thf(fact_706_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_707_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_708_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_709_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_710_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_711_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_712_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_713_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_714_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_715_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_716_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_717_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_718_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_719_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_720_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_721_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_722_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_723_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_724_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_725_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A8: A,B7: A] :
( ( ord_less @ A @ A8 @ B7 )
=> ( P @ A8 @ B7 ) )
=> ( ! [A8: A] : ( P @ A8 @ A8 )
=> ( ! [A8: A,B7: A] :
( ( P @ B7 @ A8 )
=> ( P @ A8 @ B7 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_726_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P5: A > $o] :
? [X6: A] : ( P5 @ X6 ) )
= ( ^ [P2: A > $o] :
? [N2: A] :
( ( P2 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P2 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_727_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_728_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_729_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_730_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_731_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_732_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_733_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_734_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_735_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_736_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_737_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A8: A,B7: A] :
( ( ord_less_eq @ A @ A8 @ B7 )
=> ( P @ A8 @ B7 ) )
=> ( ! [A8: A,B7: A] :
( ( P @ B7 @ A8 )
=> ( P @ A8 @ B7 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_738_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_739_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X2: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X2 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_740_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_741_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_742_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_743_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_744_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_745_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_746_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_747_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_748_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_749_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_750_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_751_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_752_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_753_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_754_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( B2 != A3 ) ) ) ) ).
% nle_le
thf(fact_755_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% nless_le
thf(fact_756_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_757_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_758_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_759_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_760_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_761_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_762_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_763_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_764_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_765_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_766_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_767_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_768_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_769_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_770_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_771_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_772_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_773_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_774_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_775_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_776_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_777_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_778_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_779_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_780_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_781_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_782_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_783_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_784_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_785_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_786_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_787_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_788_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_789_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_790_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_791_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_792_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_793_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_794_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_795_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_796_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_797_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_798_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% subrelI
thf(fact_799_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_800_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_801_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_802_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_803_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_804_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_805_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_806_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_807_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_808_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_809_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_810_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_811_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_812_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_813_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_814_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_815_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_816_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_817_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_818_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_819_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_820_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_821_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_822_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_823_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_824_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_825_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_826_subset__minus__empty,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_827_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_828_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_829_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_830_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_831_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_832_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_833_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_834_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_835_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_836_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C2 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_837_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_838_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_839_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_840_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% inf.orderI
thf(fact_841_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Z3 )
=> ( ord_less_eq @ A @ X2 @ ( F2 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_842_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( inf_inf @ A @ X3 @ Y3 )
= X3 ) ) ) ) ).
% le_iff_inf
thf(fact_843_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_844_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_845_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_846_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_847_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_848_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_849_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_850_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( inf_inf @ A @ A5 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_851_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_852_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_853_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( inf_inf @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_854_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( inf_inf @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_855_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_856_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_857_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_858_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_859_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_860_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_861_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_862_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_863_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_864_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_865_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D3 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_866_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C2 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_867_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X ) ) ) ) ).
% sup_least
thf(fact_868_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( sup_sup @ A @ X3 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_869_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_870_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_871_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ( ord_less_eq @ A @ ( F2 @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_872_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_873_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_874_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_875_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_876_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_877_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_878_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_879_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_880_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_881_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( sup_sup @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_882_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( sup_sup @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_883_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_884_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_885_subset__UNIV,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_886_inter__eq__subsetI,axiom,
! [A: $tType,S: set @ A,S4: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ S4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ S4 )
= ( inf_inf @ ( set @ A ) @ B3 @ S4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ S )
= ( inf_inf @ ( set @ A ) @ B3 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_887_Int__Collect__mono,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_888_Int__greatest,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_889_Int__absorb2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= A4 ) ) ).
% Int_absorb2
thf(fact_890_Int__absorb1,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_891_Int__lower2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_892_Int__lower1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ A4 ) ).
% Int_lower1
thf(fact_893_Int__mono,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B3: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_894_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_895_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ A4 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B3 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_896_Un__absorb2,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= A4 ) ) ).
% Un_absorb2
thf(fact_897_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_898_Un__upper2,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_upper2
thf(fact_899_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_upper1
thf(fact_900_Un__least,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C3 ) ) ) ).
% Un_least
thf(fact_901_Un__mono,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B3: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_902_Diff__partition,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
= B3 ) ) ).
% Diff_partition
thf(fact_903_Diff__subset__conv,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_904_relH__subset,axiom,
! [Bs: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,As: set @ nat] :
( ( relH @ Bs @ H3 @ H4 )
=> ( ( ord_less_eq @ ( set @ nat ) @ As @ Bs )
=> ( relH @ As @ H3 @ H4 ) ) ) ).
% relH_subset
thf(fact_905_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_906_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_907_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_908_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_909_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_910_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_911_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_912_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_913_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_914_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_915_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_916_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_917_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_918_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_919_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_920_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_921_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_922_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_923_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_924_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_925_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_926_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_927_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_928_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_929_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_930_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_931_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_932_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_933_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_934_less__eq__assn__def,axiom,
( ( ord_less_eq @ assn )
= ( ^ [A5: assn,B4: assn] :
( A5
= ( inf_inf @ assn @ A5 @ B4 ) ) ) ) ).
% less_eq_assn_def
thf(fact_935_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_936_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_937_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_938_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_939_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_940_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_941_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_942_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_943_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_944_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_945_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_946_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_947_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_948_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_949_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_950_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_951_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_952_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_953_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_954_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_955_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_956_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_957_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_958_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_959_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_960_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_961_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_962_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_963_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_964_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_965_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_966_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_967_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_968_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_969_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_970_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_971_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_972_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_973_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_974_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_975_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_976_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_977_zero__less__one__class_Ozero__le__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one_class.zero_le_one
thf(fact_978_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_979_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_980_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_981_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_982_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_983_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(4)
thf(fact_984_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% less_minus_one_simps(2)
thf(fact_985_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_986_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_987_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_988_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_989_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(4)
thf(fact_990_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% le_minus_one_simps(2)
thf(fact_991_disjoint__alt__simp3,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ A4 )
= ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_992_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_993_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_994_disjoint__mono,axiom,
! [A: $tType,A3: set @ A,A7: set @ A,B2: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A7 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_995_Un__Int__assoc__eq,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C3 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_996_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_997_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_998_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_999_in__range__subset,axiom,
! [As: set @ nat,As3: set @ nat,H3: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As @ As3 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ).
% in_range_subset
thf(fact_1000_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1001_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_1002_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(3)
thf(fact_1003_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% less_minus_one_simps(1)
thf(fact_1004_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1005_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1006_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1007_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1008_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1009_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1010_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1011_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X )
=> ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1012_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1013_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1014_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1015_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1016_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_1017_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_less_numeral
thf(fact_1018_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_1019_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1020_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1021_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1022_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1023_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1024_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_1025_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1026_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_1027_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(3)
thf(fact_1028_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% le_minus_one_simps(1)
thf(fact_1029_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1030_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1031_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_le_numeral
thf(fact_1032_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1033_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1034_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1035_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_1036_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_1037_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_1038_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% shunt2
thf(fact_1039_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P4: A,Q4: A,R3: A] :
( ( ord_less_eq @ A @ P4 @ ( sup_sup @ A @ Q4 @ R3 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P4 @ ( uminus_uminus @ A @ Q4 ) ) @ R3 ) ) ) ).
% sup_neg_inf
thf(fact_1040_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1041_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1042_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1043_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 ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1044_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1045_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1046_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1047_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1048_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1049_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1050_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 ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1051_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1052_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X ) @ ( times_times @ A @ Z2 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1053_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1054_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1055_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
thf(fact_1056_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_1057_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1058_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
= ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1059_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_1060_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_1061_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_1062_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_1063_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q2: A > B > $o] :
( ! [X2: A,Y2: B] :
( ( P @ X2 @ Y2 )
=> ( Q2 @ X2 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 ) ) ).
% predicate2I
thf(fact_1064_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_1065_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_1066_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_1067_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_1068_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_1069_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_1070_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ B2 @ A3 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_1071_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_1072_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1073_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1074_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_1075_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_1076_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_1077_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_1078_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_1079_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_1080_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_1081_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_1082_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_1083_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_1084_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_1085_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_1086_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_1087_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_1088_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_1089_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_1090_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1091_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1092_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_1093_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_1094_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_1095_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_1096_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1097_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1098_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1099_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1100_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1101_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_1102_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_1103_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1104_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1105_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_1106_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_1107_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1108_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1109_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_1110_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_1111_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(7)
thf(fact_1112_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(8)
thf(fact_1113_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1114_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1115_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1116_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1117_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_1118_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_1119_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_1120_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_1121_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_1122_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1123_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_1124_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_1125_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1126_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1127_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_1128_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A5: assn,B4: assn] :
( ( ord_less_eq @ assn @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% less_assn_def
thf(fact_1129_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q2: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
=> ( Q2 @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_1130_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
& ~ ( ord_less_eq @ ( A > B ) @ G @ F ) ) ) ) ) ).
% less_fun_def
thf(fact_1131_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q2: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
=> ( ( P @ X @ Y )
=> ( Q2 @ X @ Y ) ) ) ).
% predicate2D
thf(fact_1132_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [A5: A,B4: A] :
( ( P @ A5 @ B4 )
| ( A5 = B4 ) ) ) ).
% eq_subset
thf(fact_1133_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_1134_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_1135_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_1136_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_1137_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_1138_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_1139_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_1140_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( ( ( F2 @ A3 @ B2 )
= Z2 )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_1141_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( ( Z2
= ( F2 @ A3 @ B2 ) )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_1142_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 ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1143_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_1144_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_1145_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_1146_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_1147_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_1148_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_1149_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_1150_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_1151_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_1152_add_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.right_assoc
thf(fact_1153_add_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% add.right_commute
thf(fact_1154_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_1155_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1156_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1157_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1158_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_1159_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_1160_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A3 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_1161_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_1162_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C5: A] :
( B4
= ( plus_plus @ A @ A5 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_1163_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_1164_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_1165_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1166_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_1167_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_1168_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1169_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1170_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1171_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1172_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_1173_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_1174_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_1175_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_1176_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E4: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E4 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_1177_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_1178_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_1179_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1180_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1181_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1182_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1183_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_1184_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( A3
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_1185_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_1186_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_1187_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_1188_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1189_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A3 )
=> ( C2
= ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_1190_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_1191_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_1192_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1193_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1194_add_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A3 )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% add.safe_commute
thf(fact_1195_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1196_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1197_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1198_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1199_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_1200_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_1201_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_1202_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_1203_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1204_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1205_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1206_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1207_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1208_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1209_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1210_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1211_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1212_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1213_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1214_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1215_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1216_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1217_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1218_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1219_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1220_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1221_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1222_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1223_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_1224_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_1225_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1226_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1227_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1228_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1229_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_1230_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1231_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_1232_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_1233_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_1234_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ D3 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E4 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1235_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E4 ) @ C2 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1236_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1237_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1238_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1239_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_1240_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_1241_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_1242_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_1243_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_1244_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.comm_monoid_axioms
thf(fact_1245_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_1246_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( semilattice_neutr @ A @ F2 @ Z2 ) ) ).
% semilattice_neutr_order.axioms(1)
thf(fact_1247_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_1248_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1249_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1250_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1251_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1252_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1253_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1254_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1255_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1256_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1257_zero__less__two,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_1258_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E4 ) @ C2 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1259_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E4 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1260_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E4 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1261_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E4 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E4 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1262_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1263_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1264_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1265_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1266_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1267_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1268_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1269_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_1270_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1271_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1272_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1273_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1274_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_1275_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_1276_uminus__numeral__One,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% uminus_numeral_One
thf(fact_1277_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_1278_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_1279_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_1280_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_1281_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_1282_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X3: A] : ( minus_minus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1283_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_1284_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1285_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1286_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1287_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1288_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R3: A,S3: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1289_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
thf(fact_1290_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_1291_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_1292_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1293_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1294_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A3: A,B2: A,C2: A,D3: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R3 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R3 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1295_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1296_less__by__empty,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( A4
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B3 ) ) ).
% less_by_empty
thf(fact_1297_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num] :
( ( unique8689654367752047608divmod @ A @ M @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1298_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_1299_predicate1I,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q2 ) ) ).
% predicate1I
thf(fact_1300_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W2 ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_1301_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_1302_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_1303_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_1304_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% diff_numeral_simps(4)
thf(fact_1305_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_1306_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% diff_numeral_special(1)
thf(fact_1307_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_1308_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_1309_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_1310_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_1311_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% diff_numeral_special(8)
thf(fact_1312_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_1313_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N )
= M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_1314_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1315_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_1316_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q2 )
=> ( Q2 @ X ) ) ) ).
% rev_predicate1D
thf(fact_1317_predicate1D,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q2 )
=> ( ( P @ X )
=> ( Q2 @ X ) ) ) ).
% predicate1D
thf(fact_1318_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y @ X ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_1319_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_1320_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_1321_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_1322_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( A3 != B2 )
& ( C2 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1323_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_1324_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1325_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1326_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X2: A] :
~ ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_1327_dbl__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% dbl_simps(4)
thf(fact_1328_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_1329_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_1330_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R3: A,Q4: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R3 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R3 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R3 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1331_one__add__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% one_add_one
thf(fact_1332_dbl__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% dbl_simps(3)
thf(fact_1333_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% dbl_inc_simps(3)
thf(fact_1334_one__div__two__eq__zero,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% one_div_two_eq_zero
thf(fact_1335_bits__1__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% bits_1_div_2
thf(fact_1336_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_1337_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_1338_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% diff_numeral_special(11)
thf(fact_1339_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_1340_sub__num__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% sub_num_simps(3)
thf(fact_1341_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_1342_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_1343_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq @ nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less @ nat @ Y3 @ X3 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1344_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_1345_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M3: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M3: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_1346_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1347_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1348_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1349_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_code(2)
thf(fact_1350_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_1351_nat__1__add__1,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% nat_1_add_1
thf(fact_1352_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S3: B,R: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1353_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_1354_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1355_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1356_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1357_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1358_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_1359_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2
thf(fact_1360_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2_right
thf(fact_1361_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_1362_prop__restrict,axiom,
! [A: $tType,X: A,Z6: set @ A,X7: set @ A,P: A > $o] :
( ( member @ A @ X @ Z6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z6
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1363_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_1364_minus__1__div__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% minus_1_div_2_eq
thf(fact_1365_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_1366_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1367_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_1368_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_1369_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R4 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_1370_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1371_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_1372_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_minus_minus
thf(fact_1373_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A3: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F2 @ A3 @ B2 ) ) ).
% case_prod_conv
thf(fact_1374_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1375_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1376_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
thf(fact_1377_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1378_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1379_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1380_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1381_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1382_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A3 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_mod_self1
thf(fact_1383_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_1384_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_one_mod_two_eq_one
thf(fact_1385_one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_mod_two_eq_one
thf(fact_1386_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_1387_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% xor_numerals(8)
thf(fact_1388_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% xor_numerals(5)
thf(fact_1389_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_1390_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_1391_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1392_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1393_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% minus_1_mod_2_eq
thf(fact_1394_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_minus_1_mod_2_eq
thf(fact_1395_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_1396_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_1397_zmod__numeral__Bit1,axiom,
! [V: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_1398_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_1399_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_1400_zmod__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_1401_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_1402_zdiv__zminus1__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_1403_zdiv__zminus2__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_1404_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1405_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1406_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1407_verit__less__mono__div__int2,axiom,
! [A4: int,B3: int,N: int] :
( ( ord_less_eq @ int @ A4 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B3 @ N ) @ ( divide_divide @ int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1408_verit__le__mono__div__int,axiom,
! [A4: int,B3: int,N: int] :
( ( ord_less @ int @ A4 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B3 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B3 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1409_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_1410_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M2: num,N2: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_1411_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_1412_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_1413_zmod__zminus2__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_1414_zmod__zminus1__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_1415_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H3: C > D,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H3 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X23: B] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1416_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [L2: B,R4: C] : ( F @ L2 @ R4 @ Y3 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_1417_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [A5: B,B4: C] : ( F @ A5 @ B4 @ Y3 )
@ X3 ) ) ) ).
% nested_case_prod_simp
thf(fact_1418_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_1419_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A7: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A7 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_1420_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_1421_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_1422_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_1423_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_1424_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% mod_minus_eq
thf(fact_1425_mod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,A7: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( modulo_modulo @ A @ A7 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A7 ) @ B2 ) ) ) ) ).
% mod_minus_cong
thf(fact_1426_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% mod_minus_right
thf(fact_1427_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F2: A > B > C,G2: A > B > C,P4: product_prod @ A @ B] :
( ! [X2: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X2 @ Y2 )
= Q4 )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P4 )
= ( product_case_prod @ A @ B @ C @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_1428_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_1429_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_1430_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X2: B,Y2: C] :
( ( Z2
= ( product_Pair @ B @ C @ X2 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X2 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_1431_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( F2 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_1432_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G2: ( product_prod @ A @ B ) > C] :
( ! [X2: A,Y2: B] :
( ( F2 @ X2 @ Y2 )
= ( G2 @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_1433_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
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_1434_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_1435_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_1436_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M2: num,N2: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_1437_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R4: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R4 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R4 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_1438_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_1439_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.comm_monoid_axioms
thf(fact_1440_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_1441_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_1442_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_1443_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_1444_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_1445_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1446_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1447_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1448_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1449_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1450_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1451_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1452_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1453_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N2: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ) ).
% divmod_def
thf(fact_1454_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ 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
thf(fact_1455_verit__le__mono__div,axiom,
! [A4: nat,B3: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B3 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B3 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1456_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R4: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R4 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R4 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_1457_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_1458_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_1459_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1460_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_1461_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1462_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1463_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_1464_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_1465_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1466_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1467_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_1468_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_1469_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A8: A,B7: B] :
( ( P4
= ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( C2 @ A8 @ B7 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_1470_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A3: A,B2: B] :
( ( F2 @ A3 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% case_prodI
thf(fact_1471_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A8: A,B7: B] :
( ( P4
= ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( member @ C @ Z2 @ ( C2 @ A8 @ B7 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_1472_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A3 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1473_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A8: A,B7: B] :
( ( ( product_Pair @ A @ B @ A8 @ B7 )
= P4 )
=> ( C2 @ A8 @ B7 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X ) ) ).
% case_prodI2'
thf(fact_1474_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_1475_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_1476_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_1477_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_1478_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
=> ~ ! [X2: B,Y2: C] :
( ( P4
= ( product_Pair @ B @ C @ X2 @ Y2 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_1479_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_1480_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X2: A,Y2: B] :
( ( P4
= ( product_Pair @ A @ B @ X2 @ Y2 ) )
=> ~ ( C2 @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_1481_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A3: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( F2 @ A3 @ B2 ) ) ).
% case_prodD
thf(fact_1482_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z2 )
=> ~ ! [X2: A,Y2: B] :
( ( P4
= ( product_Pair @ A @ B @ X2 @ Y2 ) )
=> ~ ( C2 @ X2 @ Y2 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_1483_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C2 )
=> ( R @ A3 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_1484_Id__on__def_H,axiom,
! [A: $tType,A4: A > $o] :
( ( id_on @ A @ ( collect @ A @ A4 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( X3 = Y3 )
& ( A4 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_1485_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V2: A,W3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W3 ) @ R2 )
& ~ ( member @ A @ V2 @ A6 )
& ~ ( member @ A @ W3 @ A6 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_1486_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1487_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_1488_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_1489_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1490_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1491_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1492_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1493_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1494_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1495_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1496_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ Z2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1497_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1498_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 )
= ( ord_less @ int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1499_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1500_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1501_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1502_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_1503_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_1504_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_1505_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_1506_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_1507_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] :
( P
& ( Q2 @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_1508_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_1509_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_1510_Suc__diff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_1511_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_1512_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1513_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1514_sub__num__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% sub_num_simps(2)
thf(fact_1515_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_1516_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_1517_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1518_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_1519_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_1520_unique__remainder,axiom,
! [A3: int,B2: int,Q4: int,R3: int,Q6: int,R5: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R5 ) )
=> ( R3 = R5 ) ) ) ).
% unique_remainder
thf(fact_1521_unique__quotient,axiom,
! [A3: int,B2: int,Q4: int,R3: int,Q6: int,R5: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R5 ) )
=> ( Q4 = Q6 ) ) ) ).
% unique_quotient
thf(fact_1522_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1523_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1524_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1525_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1526_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1527_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_1528_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1529_div__int__unique,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( divide_divide @ int @ K @ L )
= Q4 ) ) ).
% div_int_unique
thf(fact_1530_mod__int__unique,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R3 ) ) ).
% mod_int_unique
thf(fact_1531_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K4: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K4 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_1532_small__lazy_H_Ocases,axiom,
! [X: product_prod @ int @ int] :
~ ! [D2: int,I2: int] :
( X
!= ( product_Pair @ int @ int @ D2 @ I2 ) ) ).
% small_lazy'.cases
thf(fact_1533_exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F3: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
( X
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F3 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_1534_full__exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F3: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
( X
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F3 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_1535_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1536_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1537_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1538_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1539_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q4 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1540_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_1541_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_1542_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1543_zminus1__lemma,axiom,
! [A3: int,B2: int,Q4: int,R3: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q4 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_1544_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1545_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1546_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1547_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R3 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
& ( ord_less @ int @ R3 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R3 )
& ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q4
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1548_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M2 @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_1549_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1550_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_1551_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z5: int] :
( ( ord_less_eq @ int @ D5 @ Z5 )
& ( ord_less @ int @ Z7 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_1552_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z5: int] :
( ( ord_less_eq @ int @ D5 @ Z7 )
& ( ord_less @ int @ Z7 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_1553_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_1554_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_1555_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
thf(fact_1556_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% take_bit_minus
thf(fact_1557_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_1558_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F3: nat > A > A,A8: nat,B7: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A8 @ ( product_Pair @ nat @ A @ B7 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1559_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_1560_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_1561_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_1562_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_1563_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_1564_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_1565_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_1566_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_1567_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_1568_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_1569_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_1570_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
=> ( ! [F3: nat > A > A,A8: nat,B7: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A8 @ ( product_Pair @ nat @ A @ B7 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B7 @ A8 )
=> ( P @ F3 @ ( plus_plus @ nat @ A8 @ ( one_one @ nat ) ) @ B7 @ ( F3 @ A8 @ Acc ) ) )
=> ( P @ F3 @ A8 @ B7 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_1571_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K4: int,L2: int] :
( if @ int
@ ( ( K4
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K4
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K4
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_1572_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_1573_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_1574_normalize__negative,axiom,
! [Q4: int,P4: int] :
( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_1575_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1576_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1577_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_1578_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_1579_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_1580_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_1581_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
= ( M
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_1582_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1583_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1584_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1585_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1586_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1587_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1588_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 @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1589_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M )
= ( power_power @ A @ A3 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_1590_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_1591_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1592_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
thf(fact_1593_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
thf(fact_1594_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
thf(fact_1595_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_1596_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_1597_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X ) ) ).
% bit.conj_one_right
thf(fact_1598_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1599_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1600_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1601_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_1602_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_1603_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_1604_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_1605_mask__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% mask_Suc_0
thf(fact_1606_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1607_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1608_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_1609_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_1610_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1611_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_1612_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1613_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_1614_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_1615_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_1616_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_1617_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_1618_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_1619_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1620_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1621_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_1622_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_1623_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_1624_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_1625_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_1626_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_1627_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_1628_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1629_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1630_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1631_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_1632_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_1633_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_1634_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1635_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_1636_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1637_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1638_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1639_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1640_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1641_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_commutes
thf(fact_1642_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ~ ! [K2: A] :
( A3
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_1643_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B2: A,K: A] :
( ( A3
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% dvdI
thf(fact_1644_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B4: A,A5: A] :
? [K4: A] :
( A5
= ( times_times @ A @ B4 @ K4 ) ) ) ) ) ).
% dvd_def
thf(fact_1645_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1646_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1647_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1648_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_1649_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 @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1650_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1651_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_1652_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_1653_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
thf(fact_1654_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
thf(fact_1655_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D5 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_1656_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int,T3: int] : ( dvd_dvd @ int @ D5 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1657_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1658_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_1659_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A3 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_1660_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A3 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1661_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_1662_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_1663_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_1664_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_1665_not__is__unit__0,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% not_is_unit_0
thf(fact_1666_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1667_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_1668_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_1669_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ 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
thf(fact_1670_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 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1671_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 @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1672_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 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1673_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 @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1674_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1675_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1676_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1677_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_1678_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_1679_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_1680_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 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1681_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1682_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1683_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 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1684_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_1685_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
thf(fact_1686_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
thf(fact_1687_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
thf(fact_1688_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_1689_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_1690_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_1691_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N2: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_1692_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N2: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_1693_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1694_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1695_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1696_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X3: A] : ( P @ ( times_times @ A @ L @ X3 ) ) )
= ( ? [X3: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X3 @ ( zero_zero @ A ) ) )
& ( P @ X3 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1697_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_1698_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B2
!= ( times_times @ A @ A3 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1699_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1700_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.comm_monoid_axioms
thf(fact_1701_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1702_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N4 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1703_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N4 ) ) ) ) ) ).
% power_increasing
thf(fact_1704_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_1705_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T4: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X5: A,K3: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X5 @ T4 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K3 @ D4 ) ) @ T4 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1706_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T4: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X5: A,K3: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X5 @ T4 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K3 @ D4 ) ) @ T4 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1707_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 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A3 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1708_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 @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1709_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 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1710_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
= ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1711_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
thf(fact_1712_and_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.monoid_axioms
thf(fact_1713_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
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1714_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 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_1715_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 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1716_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1717_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1718_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 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1719_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
thf(fact_1720_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_1721_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.semilattice_neutr_axioms
thf(fact_1722_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_1723_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_1724_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W2: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit0 @ W2 ) ) )
= ( times_times @ A @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_even
thf(fact_1725_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W2: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit1 @ W2 ) ) )
= ( times_times @ A @ ( times_times @ A @ Z2 @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_odd
thf(fact_1726_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_1727_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_1728_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1729_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1730_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1731_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N4 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1732_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N4 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1733_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1734_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1735_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1736_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 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1737_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 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1738_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B7: A] :
( ( B7
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B7 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B7 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B7 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B7 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1739_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B7: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B7 ) ) ) ) ).
% evenE
thf(fact_1740_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1741_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_minus
thf(fact_1742_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A] :
( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% power4_eq_xxxx
thf(fact_1743_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_1744_one__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_power2
thf(fact_1745_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% power2_eq_iff
thf(fact_1746_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_1747_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_1748_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_1749_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1750_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_1751_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_1752_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_1753_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_1754_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_1755_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_1756_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1757_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_1758_normalize__denom__pos,axiom,
! [R3: product_prod @ int @ int,P4: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_1759_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1760_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1761_normalize__crossproduct,axiom,
! [Q4: int,S3: int,P4: int,R3: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S3
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R3 @ S3 ) ) )
=> ( ( times_times @ int @ P4 @ S3 )
= ( times_times @ int @ R3 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_1762_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1763_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_1764_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B7: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B7 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1765_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_1766_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_1767_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_1768_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_1769_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_1770_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_1771_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_1772_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_1773_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_1774_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_1775_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_1776_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1777_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_1778_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_1779_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_1780_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_1781_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_1782_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_1783_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_1784_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A3: nat,B2: nat,Acc2: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F2 @ A3 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_1785_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_1786_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1787_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_1788_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_1789_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_1790_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_1791_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_1792_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_1793_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
= ~ P ) ) ).
% of_bool_less_one_iff
thf(fact_1794_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_1795_of__int__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( one_one @ int ) )
= ( one_one @ A ) ) ) ).
% of_int_1
thf(fact_1796_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( one_one @ A ) )
= ( Z2
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_1797_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z2 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_mult
thf(fact_1798_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_minus
thf(fact_1799_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R3: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
= ( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R3
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_1800_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_1801_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_1802_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_1803_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_le_iff
thf(fact_1804_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_1805_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_less_iff
thf(fact_1806_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_1807_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_1808_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_1809_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_1810_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_1811_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_1812_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_1813_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_1814_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_1815_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q2: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q2 ) ) ) ) ).
% of_bool_conj
thf(fact_1816_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_1817_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1818_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1819_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_1820_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R4: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R4
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_1821_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_neg_numeral
thf(fact_1822_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1823_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F: nat > A > A,A5: nat,B4: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A5 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F @ A5 @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_1824_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_1825_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_1826_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A8: A] :
( ( ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A8 )
=> ( P @ A8 ) )
=> ( ! [A8: A,B7: $o] :
( ( P @ A8 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B7 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A8 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B7 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_1827_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_1828_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P4: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A3 @ B2 ) )
=> ~ ! [X2: A,Y2: A] :
( ( P4
= ( times_times @ A @ X2 @ Y2 ) )
=> ( ( dvd_dvd @ A @ X2 @ A3 )
=> ~ ( dvd_dvd @ A @ Y2 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1829_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B9: A,C6: A] :
( ( A3
= ( times_times @ A @ B9 @ C6 ) )
& ( dvd_dvd @ A @ B9 @ B2 )
& ( dvd_dvd @ A @ C6 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_1830_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1831_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1832_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_1833_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_1834_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_1835_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
=> ( Y6 = X2 ) ) ) ) ).
% floor_exists1
thf(fact_1836_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_1837_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K4: int,L2: int] :
( if @ int
@ ( ( member @ int @ K4 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_1838_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_1839_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_1840_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_1841_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R4: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R4 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R4 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_1842_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_1843_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1844_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1845_insert__inter__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ ( insert2 @ A @ A3 @ B3 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1846_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C3: set @ A,B3: set @ A] :
( ( member @ A @ A3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B3 ) @ C3 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1847_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C3: set @ A,B3: set @ A] :
( ~ ( member @ A @ A3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1848_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A3: A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_1849_Un__insert__left,axiom,
! [A: $tType,A3: A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ B3 ) @ C3 )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1850_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ A3 ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_1851_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X3: A] : X3 = A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_1852_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A4: set @ A] :
( ( ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ A3 @ A4 ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1853_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: A] :
( ( ( insert2 @ A @ A3 @ A4 )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1854_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_1855_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ B3 ) )
= ( ~ ( member @ A @ A3 @ B3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1856_disjoint__insert_I1_J,axiom,
! [A: $tType,B3: set @ A,A3: A,A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B3 @ ( insert2 @ A @ A3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ B3 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_1857_disjoint__insert_I2_J,axiom,
! [A: $tType,A4: set @ A,B2: A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ B3 ) ) )
= ( ~ ( member @ A @ B2 @ A4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1858_insert__Diff__single,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert2 @ A @ A3 @ A4 ) ) ).
% insert_Diff_single
thf(fact_1859_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_1860_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_1861_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_1862_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% round_neg_numeral
thf(fact_1863_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_1864_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_1865_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_1866_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_1867_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert2 @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A3 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1868_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( X3 = A5 )
| ( member @ A @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1869_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M2: num,N2: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N2 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_1870_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F3: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
( X
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F3 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_1871_full__exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F3: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
( X
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F3 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_1872_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya: A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( insert2 @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
= ( bNF_id_bnf @ ( A > $o ) @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_1873_ID_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: A,Pa: A > $o] :
( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( bNF_id_bnf @ ( A > $o ) @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_1874_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya: A,Y: B,Xa: B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( insert2 @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member @ B @ Yb @ ( insert2 @ B @ Xa @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_id_bnf @ ( A > B > $o ) @ R @ X @ Y )
= ( bNF_id_bnf @ ( A > B > $o ) @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_1875_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: A,Y: B,Ra: A > B > $o] :
( ( bNF_id_bnf @ ( A > B > $o ) @ R @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member @ B @ Yb @ ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_id_bnf @ ( A > B > $o ) @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_1876_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_id_bnf @ ( A > A > $o ) @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_1877_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_1878_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert2 @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_1879_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C2: A,D3: A] :
( ( ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C2 @ ( insert2 @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B2 = D3 ) )
| ( ( A3 = D3 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1880_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_1881_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_1882_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N2: nat,K4: int] : ( bit_se5824344971392196577ns_xor @ int @ K4 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_1883_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert2 @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_1884_Int__insert__right,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1885_Int__insert__left,axiom,
! [A: $tType,A3: A,C3: set @ A,B3: set @ A] :
( ( ( member @ A @ A3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B3 ) @ C3 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) )
& ( ~ ( member @ A @ A3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1886_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M2: nat,N2: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_1887_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) ) ) ) ).
% push_bit_minus
thf(fact_1888_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_1889_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_1890_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] : X3 = A5 ) ) ) ) ).
% insert_def
thf(fact_1891_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_1892_subset__singleton__iff,axiom,
! [A: $tType,X7: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
| ( X7
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_1893_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B3: set @ A] :
( ( ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1894_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1895_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_1896_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X7: set @ A] :
( ~ ( member @ A @ X @ X7 )
=> ( ( minus_minus @ ( set @ A ) @ X7 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_1897_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_1898_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_1899_insert__Diff,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1900_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A3: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1901_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1902_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B2: B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F2 @ A3 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_1903_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B3 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1904_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1905_remove__subset,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S ) ) ).
% remove_subset
thf(fact_1906_vimage__insert,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: B,B3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ).
% vimage_insert
thf(fact_1907_Compl__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_1908_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_1909_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_1910_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_1911_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B3 ) )
= ( ( ( member @ A @ X @ B3 )
=> ( ord_less @ ( set @ A ) @ A4 @ B3 ) )
& ( ~ ( member @ A @ X @ B3 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1912_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_1913_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N2: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_1914_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_1915_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_1916_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_1917_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_1918_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K4: int] :
( if @ code_integer @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K4 ) ) )
@ ( if @ code_integer
@ ( K4
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_1919_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_1920_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_1921_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1922_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1923_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1924_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_1925_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_1926_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1927_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( uminus_uminus @ int @ Z2 ) ) ) ).
% floor_uminus_of_int
thf(fact_1928_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_1929_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_1930_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_1931_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1932_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_1933_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_1934_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1935_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_1936_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_neg_numeral
thf(fact_1937_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_1938_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_1939_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_1940_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_1941_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_1942_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_1943_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_1944_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1945_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1946_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_1947_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_1948_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_1949_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_1950_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_1951_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_1952_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_1953_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y2: int] :
( ( X
= ( code_integer_of_int @ Y2 ) )
=> ~ ( member @ int @ Y2 @ ( top_top @ ( set @ int ) ) ) ) ).
% integer_of_int_cases
thf(fact_1954_integer__of__int__induct,axiom,
! [P: code_integer > $o,X: code_integer] :
( ! [Y2: int] :
( ( member @ int @ Y2 @ ( top_top @ ( set @ int ) ) )
=> ( P @ ( code_integer_of_int @ Y2 ) ) )
=> ( P @ X ) ) ).
% integer_of_int_induct
thf(fact_1955_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
=> ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( ( code_integer_of_int @ X )
= ( code_integer_of_int @ Y ) )
= ( X = Y ) ) ) ) ).
% integer_of_int_inject
thf(fact_1956_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_1957_ID_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: A,Ya: A,F2: A > B,G2: A > B] :
( ( X = Ya )
=> ( ( bNF_id_bnf @ ( A > $o )
@ ^ [Z5: A] :
( ( F2 @ Z5 )
= ( G2 @ Z5 ) )
@ Ya )
=> ( ( bNF_id_bnf @ ( A > B ) @ F2 @ X )
= ( bNF_id_bnf @ ( A > B ) @ G2 @ Ya ) ) ) ) ).
% ID.map_cong_pred
thf(fact_1958_ID_Opred__True,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o )
@ ^ [Uu: A] : $true )
= ( ^ [Uu: A] : $true ) ) ).
% ID.pred_True
thf(fact_1959_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X3: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ) ) ).
% ceiling_def
thf(fact_1960_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% floor_minus
thf(fact_1961_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% ceiling_minus
thf(fact_1962_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_1963_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_1964_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X3: A] :
( if @ int
@ ( X3
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
@ ( archim6421214686448440834_floor @ A @ X3 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_1965_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_1966_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_1967_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_1968_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1969_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T4: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T4 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T4 )
& ( ord_less @ A @ T4 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_1970_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_1971_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_1972_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_1973_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_1974_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_1975_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_1976_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1977_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T4: A] :
( ( P @ ( archimedean_ceiling @ A @ T4 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T4 )
& ( ord_less_eq @ A @ T4 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_1978_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1979_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1980_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_1981_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_1982_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1983_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_1984_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_1985_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_1986_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K2: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_1987_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1988_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X3 ) ) @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ).
% round_altdef
thf(fact_1989_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_1990_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_1991_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_1992_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K4: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K4 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L2: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
@ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_1993_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K4 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_1994_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_1995_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1996_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_1997_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_1998_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_1999_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_2000_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2001_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus
thf(fact_2002_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_2003_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2004_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_2005_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_2006_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_2007_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_2008_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_2009_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_2010_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_2011_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z2 ) @ ( one_one @ int ) )
= ( Z2
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_2012_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_2013_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_2014_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_2015_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_2016_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2017_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_2018_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_2019_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_2020_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_2021_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_2022_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_2023_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_2024_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_2025_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_2026_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2027_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_2028_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_2029_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_2030_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_2031_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less @ A @ A3 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_2032_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_2033_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_2034_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_1_eq
thf(fact_2035_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2036_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_2037_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2038_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_2039_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B2 = A3 )
| ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_2040_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( abs_abs @ A @ A3 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_2041_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2042_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_2043_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if_raw
thf(fact_2044_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_2045_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_2046_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_2047_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K4: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K4 ) @ K4 ) ) ) ).
% abs_integer_code
thf(fact_2048_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2049_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2050_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2051_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_2052_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2053_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_2054_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_2055_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2056_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2057_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2058_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_2059_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_2060_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2061_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_2062_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2063_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K4 @ L2 ) @ ( modulo_modulo @ code_integer @ K4 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_2064_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R4: code_integer,S2: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R4 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ S2 ) )
@ ( S2
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_2065_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K4: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L2: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_2066_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K4: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K4 ) ) )
@ ( if @ int
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_2067_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K4: int,L2: int] :
( if @ int
@ ( K4
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K4 )
@ ( if @ int
@ ( K4
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K4
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_2068_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert2 @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_2069_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% integer_of_num(3)
thf(fact_2070_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2071_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( ring_1_of_int @ A @ ( F2 @ X3 ) )
@ A4 ) ) ) ).
% of_int_sum
thf(fact_2072_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_2073_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
= ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_2074_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_2075_bit_Ocompl__zero,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.compl_zero
thf(fact_2076_bit_Ocompl__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% bit.compl_one
thf(fact_2077_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_2078_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_2079_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_right
thf(fact_2080_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_left
thf(fact_2081_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_2082_not__minus__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% not_minus_numeral_eq
thf(fact_2083_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_2084_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se5824344971392196577ns_xor @ int @ K @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_2085_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_2086_not__one__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% not_one_eq
thf(fact_2087_int__of__integer,axiom,
! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% int_of_integer
thf(fact_2088_int__of__integer__cases,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ~ ! [X2: code_integer] :
( Y
!= ( code_int_of_integer @ X2 ) ) ) ).
% int_of_integer_cases
thf(fact_2089_int__of__integer__induct,axiom,
! [Y: int,P: int > $o] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ! [X2: code_integer] : ( P @ ( code_int_of_integer @ X2 ) )
=> ( P @ Y ) ) ) ).
% int_of_integer_induct
thf(fact_2090_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A3: A,A4: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( modulo_modulo @ A @ ( F2 @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ A3 ) ) ) ).
% mod_sum_eq
thf(fact_2091_type__definition__integer,axiom,
type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% type_definition_integer
thf(fact_2092_integer__of__int__inverse,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
= Y ) ) ).
% integer_of_int_inverse
thf(fact_2093_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_2094_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_2095_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_2096_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_2097_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K4: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K4 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_2098_and__not__numerals_I1_J,axiom,
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(1)
thf(fact_2099_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I4: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I3 ) )
@ I4 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I4 ) ) ) ) ).
% sum_power_add
thf(fact_2100_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_2101_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N2: nat,K4: int] : ( bit_se5824344872417868541ns_and @ int @ K4 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_2102_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_2103_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_2104_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_2105_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_2106_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_2107_minus__numeral__eq__not__sub__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( neg_numeral_sub @ A @ N @ one2 ) ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_2108_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_2109_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_2110_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X2: A] :
( A4
!= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_2111_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X3: A] :
( A6
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_2112_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_2113_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_2114_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_2115_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_2116_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_2117_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K4: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_2118_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2119_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_2120_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A4 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2121_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A4 ) ) ) ).
% sum_abs
thf(fact_2122_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_2123_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I4 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I2 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2124_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F2 @ A5 ) )
@ A4 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F2 @ A5 ) )
@ A4 ) ) ) ).
% abs_sum_abs
thf(fact_2125_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A4 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2126_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A4: set @ A,F2: A > nat] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_2127_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > C > A,B3: set @ C,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ I3 ) @ B3 )
@ A4 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( G2 @ I3 @ J3 )
@ A4 )
@ B3 ) ) ) ).
% sum.swap
thf(fact_2128_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F2: B > A,G2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ K5 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2129_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,H3: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ A4 ) ) ) ) ).
% sum.distrib
thf(fact_2130_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A4: set @ A,G2: C > B,B3: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ B3 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I3 ) @ ( G2 @ J3 ) )
@ B3 )
@ A4 ) ) ) ).
% sum_product
thf(fact_2131_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,A4: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ ( F2 @ N2 ) @ R3 )
@ A4 ) ) ) ).
% sum_distrib_right
thf(fact_2132_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F2: B > A,A4: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ R3 @ ( F2 @ N2 ) )
@ A4 ) ) ) ).
% sum_distrib_left
thf(fact_2133_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G2: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ).
% sum_subtractf
thf(fact_2134_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( uminus_uminus @ A @ ( F2 @ X3 ) )
@ A4 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_negf
thf(fact_2135_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,A4: set @ B,R3: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( divide_divide @ A @ ( F2 @ N2 ) @ R3 )
@ A4 ) ) ) ).
% sum_divide_distrib
thf(fact_2136_sum__subtractf__nat,axiom,
! [A: $tType,A4: set @ A,G2: A > nat,F2: A > nat] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ nat @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( minus_minus @ nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G2 @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2137_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ ( code_divmod_abs @ K4 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S2 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K4 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S2 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_2138_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_2139_arith__series__nat,axiom,
! [A3: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I3 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2140_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_2141_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_2142_sum__power2,axiom,
! [K: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_2143_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_2144_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_2145_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_2146_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_2147_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_2148_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_2149_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_2150_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_2151_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_2152_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_2153_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert2 @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_2154_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_2155_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_2156_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L4: A,H4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_UNIV_eq_Icc
thf(fact_2157_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_2158_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_2159_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_2160_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_2161_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_2162_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2163_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_2164_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_2165_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_2166_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2167_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_2168_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_2169_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2170_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_2171_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2172_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
thf(fact_2173_aset_I2_J,axiom,
! [D4: int,A4: set @ int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q2 @ X5 ) )
=> ( ( P @ ( plus_plus @ int @ X5 @ D4 ) )
| ( Q2 @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_2174_aset_I1_J,axiom,
! [D4: int,A4: set @ int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q2 @ X5 ) )
=> ( ( P @ ( plus_plus @ int @ X5 @ D4 ) )
& ( Q2 @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_2175_bset_I2_J,axiom,
! [D4: int,B3: set @ int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q2 @ X5 ) )
=> ( ( P @ ( minus_minus @ int @ X5 @ D4 ) )
| ( Q2 @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_2176_bset_I1_J,axiom,
! [D4: int,B3: set @ int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q2 @ X5 ) )
=> ( ( P @ ( minus_minus @ int @ X5 @ D4 ) )
& ( Q2 @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_2177_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_2178_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_2179_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2180_aset_I10_J,axiom,
! [D3: int,D4: int,A4: set @ int,T4: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X5 @ T4 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ) ).
% aset(10)
thf(fact_2181_aset_I9_J,axiom,
! [D3: int,D4: int,A4: set @ int,T4: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X5 @ T4 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ) ).
% aset(9)
thf(fact_2182_bset_I10_J,axiom,
! [D3: int,D4: int,B3: set @ int,T4: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X5 @ T4 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ) ).
% bset(10)
thf(fact_2183_bset_I9_J,axiom,
! [D3: int,D4: int,B3: set @ int,T4: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X5 @ T4 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ) ).
% bset(9)
thf(fact_2184_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert2 @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_2185_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_2186_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_2187_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2188_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2189_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_2190_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A3: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F2 @ A5 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2191_periodic__finite__ex,axiom,
! [D3: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X4: int] : ( P @ X4 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2192_aset_I7_J,axiom,
! [D4: int,A4: set @ int,T4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T4 @ X5 )
=> ( ord_less @ int @ T4 @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_2193_aset_I5_J,axiom,
! [D4: int,T4: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T4 @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T4 )
=> ( ord_less @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ) ).
% aset(5)
thf(fact_2194_aset_I4_J,axiom,
! [D4: int,T4: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T4 @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T4 )
=> ( ( plus_plus @ int @ X5 @ D4 )
!= T4 ) ) ) ) ) ).
% aset(4)
thf(fact_2195_aset_I3_J,axiom,
! [D4: int,T4: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T4 @ ( one_one @ int ) ) @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T4 )
=> ( ( plus_plus @ int @ X5 @ D4 )
= T4 ) ) ) ) ) ).
% aset(3)
thf(fact_2196_bset_I7_J,axiom,
! [D4: int,T4: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T4 @ B3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T4 @ X5 )
=> ( ord_less @ int @ T4 @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2197_bset_I5_J,axiom,
! [D4: int,B3: set @ int,T4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T4 )
=> ( ord_less @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ).
% bset(5)
thf(fact_2198_bset_I4_J,axiom,
! [D4: int,T4: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T4 @ B3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T4 )
=> ( ( minus_minus @ int @ X5 @ D4 )
!= T4 ) ) ) ) ) ).
% bset(4)
thf(fact_2199_bset_I3_J,axiom,
! [D4: int,T4: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T4 @ ( one_one @ int ) ) @ B3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T4 )
=> ( ( minus_minus @ int @ X5 @ D4 )
= T4 ) ) ) ) ) ).
% bset(3)
thf(fact_2200_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2201_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2202_aset_I8_J,axiom,
! [D4: int,A4: set @ int,T4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T4 @ X5 )
=> ( ord_less_eq @ int @ T4 @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_2203_aset_I6_J,axiom,
! [D4: int,T4: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T4 @ ( one_one @ int ) ) @ A4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T4 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ) ).
% aset(6)
thf(fact_2204_bset_I8_J,axiom,
! [D4: int,T4: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T4 @ ( one_one @ int ) ) @ B3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T4 @ X5 )
=> ( ord_less_eq @ int @ T4 @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2205_bset_I6_J,axiom,
! [D4: int,B3: set @ int,T4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T4 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T4 ) ) ) ) ).
% bset(6)
thf(fact_2206_cpmi,axiom,
! [D4: int,P: int > $o,P7: int > $o,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z8: int] :
! [X2: int] :
( ( ord_less @ int @ X2 @ Z8 )
=> ( ( P @ X2 )
= ( P7 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( P7 @ X2 )
= ( P7 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X4: int] : ( P @ X4 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P7 @ X3 ) )
| ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ B3 )
& ( P @ ( plus_plus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2207_cppi,axiom,
! [D4: int,P: int > $o,P7: int > $o,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z8: int] :
! [X2: int] :
( ( ord_less @ int @ Z8 @ X2 )
=> ( ( P @ X2 )
= ( P7 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( P7 @ X2 )
= ( P7 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X4: int] : ( P @ X4 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P7 @ X3 ) )
| ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ A4 )
& ( P @ ( minus_minus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2208_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2209_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F2 @ K4 ) @ ( F2 @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2210_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2211_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_2212_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2213_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_2214_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2215_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2216_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_2217_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_2218_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_2219_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_2220_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_2221_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_2222_of__nat__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% of_nat_1
thf(fact_2223_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_2224_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_2225_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_2226_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( semiring_1_of_nat @ A @ ( F2 @ X3 ) )
@ A4 ) ) ) ).
% of_nat_sum
thf(fact_2227_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% of_nat_Suc
thf(fact_2228_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_2229_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_2230_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_2231_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_2232_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_2233_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_2234_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_2235_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_2236_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_2237_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2238_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2239_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_2240_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_2241_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_2242_int__Suc,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_2243_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_2244_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_2245_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_2246_int__sum,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X3: B] : ( semiring_1_of_nat @ int @ ( F2 @ X3 ) )
@ A4 ) ) ).
% int_sum
thf(fact_2247_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2248_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_2249_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_2250_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_2251_negative__zless__0,axiom,
! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_2252_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E4 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E4 ) ) ) ).
% nat_approx_posE
thf(fact_2253_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_2254_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_2255_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2256_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N2
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_2257_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2258_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2259_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ R3 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R3 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2260_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F2: nat > A,V: num,N: nat] :
( ( case_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_2261_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% gbinomial_1
thf(fact_2262_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_2263_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_2264_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_2265_case__nat__numeral,axiom,
! [A: $tType,A3: A,F2: nat > A,V: num] :
( ( case_nat @ A @ A3 @ F2 @ ( numeral_numeral @ nat @ V ) )
= ( F2 @ ( pred_numeral @ V ) ) ) ).
% case_nat_numeral
thf(fact_2266_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2267_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H3: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H3 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_2268_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H3: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2269_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H4 ) ) ) ).
% not_UNIV_eq_Iic
thf(fact_2270_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_2271_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K4 ) ) @ K4 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2272_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_2273_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_2274_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_2275_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_2276_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
thf(fact_2277_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_2278_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_2279_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_2280_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_2281_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_2282_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2283_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_2284_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A3 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2285_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K4 ) ) @ K4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K4 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_2286_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_2287_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_2288_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_2289_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2290_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2291_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_2292_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_2293_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2294_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2295_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2296_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2297_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K4: nat] :
( ( ord_less @ nat @ K4 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K4 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2298_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_2299_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_2300_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_2301_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_2302_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A5 ) @ ( one_one @ A ) ) @ K4 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_2303_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_2304_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_2305_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K4: nat] : K4
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_2306_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2307_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X3: A,F: nat > A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ X3
@ ( F @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_2308_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_2309_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_2310_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_2311_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_2312_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2313_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2314_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2315_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_2316_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R3 ) @ K ) )
= ( times_times @ A @ R3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R3 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2317_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_2318_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_2319_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_2320_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_2321_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K: nat,G2: nat > A,H3: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H3 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2322_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_2323_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_2324_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_2325_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] :
( if @ A
@ ( K4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_2326_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_2327_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2328_atMost__UNIV__triv,axiom,
! [A: $tType] :
( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atMost_UNIV_triv
thf(fact_2329_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2330_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A4 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_2331_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( semiring_1_of_nat @ A @ ( F2 @ X3 ) )
@ A4 ) ) ) ).
% of_nat_prod
thf(fact_2332_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( ring_1_of_int @ A @ ( F2 @ X3 ) )
@ A4 ) ) ) ).
% of_int_prod
thf(fact_2333_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_2334_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2335_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_2336_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2337_fact__Suc__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% fact_Suc_0
thf(fact_2338_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_2339_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_2340_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2341_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_2342_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > C > A,B3: set @ C,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G2 @ I3 ) @ B3 )
@ A4 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( G2 @ I3 @ J3 )
@ A4 )
@ B3 ) ) ) ).
% prod.swap
thf(fact_2343_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_2344_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,A4: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
!= ( one_one @ A ) )
=> ~ ! [A8: B] :
( ( member @ B @ A8 @ A4 )
=> ( ( G2 @ A8 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_2345_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2346_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,H3: B > A,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ H3 @ A4 ) ) ) ) ).
% prod.distrib
thf(fact_2347_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,G2: B > A,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ).
% prod_dividef
thf(fact_2348_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F2: A > B,A4: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ A4 ) ) ) ).
% prod_power_distrib
thf(fact_2349_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A3: A,A4: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( modulo_modulo @ A @ ( F2 @ I3 ) @ A3 )
@ A4 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ A3 ) ) ) ).
% mod_prod_eq
thf(fact_2350_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( abs_abs @ A @ ( F2 @ X3 ) )
@ A4 ) ) ) ).
% abs_prod
thf(fact_2351_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod
thf(fact_2352_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_ge_1
thf(fact_2353_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_2354_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_2355_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_2356_pochhammer__fact,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_semiring_1 @ A ) )
=> ( ( semiring_char_0_fact @ A )
= ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% pochhammer_fact
thf(fact_2357_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2358_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_2359_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F2: B > nat,A4: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A5: B] : ( power_power @ A @ C2 @ ( F2 @ A5 ) )
@ A4 ) ) ) ).
% power_sum
thf(fact_2360_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_2361_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_2362_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2363_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2364_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_2365_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X2 ) )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2366_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_2367_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P4: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_2368_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_2369_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_2370_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_2371_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_2372_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_2373_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K4 ) )
@ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_2374_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2375_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_2376_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_2377_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( binomial @ K4 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_2378_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_2379_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_2380_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_2381_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2382_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2383_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G2 @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_2384_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_2385_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2386_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2387_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2388_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_2389_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_2390_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_2391_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% fact_numeral
thf(fact_2392_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F2 @ A5 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2393_sum__choose__lower,axiom,
! [R3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( binomial @ ( plus_plus @ nat @ R3 @ K4 ) @ K4 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R3 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_2394_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_2395_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_2396_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K4: nat] : ( if @ nat @ ( ord_less @ nat @ N2 @ K4 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K4 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ K4 ) @ ( one_one @ nat ) ) @ N2 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K4 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2397_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_2398_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2399_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2400_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2401_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2402_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_2403_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( binomial @ ( minus_minus @ nat @ N @ K4 ) @ ( minus_minus @ nat @ M @ K4 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_2404_vandermonde,axiom,
! [M: nat,N: nat,R3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( times_times @ nat @ ( binomial @ M @ K4 ) @ ( binomial @ N @ ( minus_minus @ nat @ R3 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ R3 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R3 ) ) ).
% vandermonde
thf(fact_2405_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2406_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_2407_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_2408_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2409_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2410_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2411_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_2412_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_2413_binomial,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K4 ) ) @ ( power_power @ nat @ A3 @ K4 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_2414_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_2415_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_2416_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_2417_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K4 @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K4 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_2418_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_2419_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( power_power @ A @ A3 @ K4 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_2420_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2421_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K4 ) ) @ ( one_one @ A ) ) @ K4 ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_2422_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( power_power @ nat @ ( binomial @ N @ K4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_2423_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A5 ) @ K4 ) ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_2424_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G2: nat > A,H3: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H3 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2425_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2426_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_2427_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ I3 @ ( binomial @ N @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_2428_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_2429_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_2430_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F2: nat > A > A,V: num,N: nat] :
( ( rec_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_2431_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: set @ nat,C2: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D3 @ I3 ) )
@ A4 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D3 @ I3 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2432_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_2433_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_2434_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_2435_sgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% sgn_minus
thf(fact_2436_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_2437_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_2438_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2439_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_2440_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2441_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2442_prod__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( F2 @ X3 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_2443_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2444_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2445_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F2: nat > A > A,V: num] :
( ( rec_nat @ A @ A3 @ F2 @ ( numeral_numeral @ nat @ V ) )
= ( F2 @ ( pred_numeral @ V ) @ ( rec_nat @ A @ A3 @ F2 @ ( pred_numeral @ V ) ) ) ) ).
% rec_nat_numeral
thf(fact_2446_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_2447_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% prod.insert
thf(fact_2448_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_2449_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2450_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_2451_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_2452_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G2 @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_2453_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_2454_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_2455_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,F2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X3 ) ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_2456_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_2457_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_2458_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A4 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2459_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_2460_int__prod,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X3: B] : ( semiring_1_of_nat @ int @ ( F2 @ X3 ) )
@ A4 ) ) ).
% int_prod
thf(fact_2461_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set @ A,S5: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S2: set @ A] :
( ( minus_minus @ ( set @ A ) @ S2 @ S )
= ( minus_minus @ ( set @ A ) @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2462_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_2463_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2464_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less @ nat @ K4 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2465_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2466_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B3: set @ C,G2: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ X3 )
@ ( collect @ C
@ ^ [Y3: C] :
( ( member @ C @ Y3 @ B3 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( G2 @ X3 @ Y3 )
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B3 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2467_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: set @ C,G2: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G2 @ X3 )
@ ( collect @ C
@ ^ [Y3: C] :
( ( member @ C @ Y3 @ B3 )
& ( R @ X3 @ Y3 ) ) ) )
@ A4 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G2 @ X3 @ Y3 )
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( R @ X3 @ Y3 ) ) ) )
@ B3 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_2468_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_2469_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2470_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A3 ) )
=> ( ( ( sgn_sgn @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_2471_sgn__minus__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% sgn_minus_1
thf(fact_2472_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_2473_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_2474_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_2475_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K4: A] : ( times_times @ A @ K4 @ ( sgn_sgn @ A @ K4 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_2476_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
thf(fact_2477_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( X @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( Y @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2478_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( X @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2479_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G2 @ X3 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_filter
thf(fact_2480_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G2 @ X3 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_filter
thf(fact_2481_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: set @ B,F2: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2482_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_2483_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H3: B > A,G2: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X13: A,Y12: A,X24: A,Y23: A] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times @ A @ X13 @ Y12 ) @ ( times_times @ A @ X24 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H3 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2484_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_2485_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T5: set @ C,S: set @ B,I: C > B,J: B > C,T2: set @ C,G2: B > A,H3: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( ( I @ ( J @ A8 ) )
= A8 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T2 @ T5 ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T2 @ T5 ) )
=> ( ( J @ ( I @ B7 ) )
= B7 ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T2 @ T5 ) )
=> ( member @ B @ ( I @ B7 ) @ ( minus_minus @ ( set @ B ) @ S @ S4 ) ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S4 )
=> ( ( G2 @ A8 )
= ( one_one @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( H3 @ B7 )
= ( one_one @ A ) ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S )
=> ( ( H3 @ ( J @ A8 ) )
= ( G2 @ A8 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H3 @ T2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2486_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_2487_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_2488_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_2489_sum__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( F2 @ X3 )
= ( one_one @ nat ) )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F2 @ Y3 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2490_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_2491_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
thf(fact_2492_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F2: B > A,B3: A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 )
= B3 )
=> ( ( member @ B @ I @ S3 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B3 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2493_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S3 )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2494_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_2495_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(4)
thf(fact_2496_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_2497_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(2)
thf(fact_2498_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( member @ B @ X3 @ B3 ) @ ( G2 @ X3 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_restrict
thf(fact_2499_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X3: B] :
( ( G2 @ X3 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2500_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( member @ B @ X3 @ B3 ) @ ( G2 @ X3 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_restrict
thf(fact_2501_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X3: B] :
( ( G2 @ X3 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2502_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_2503_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I3: int] :
( if @ int
@ ( I3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_2504_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q2: A > nat,P: A > nat,N: A] :
( ! [X2: A] : ( ord_less_eq @ nat @ ( Q2 @ X2 ) @ ( P @ X2 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q2 @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( minus_minus @ nat @ ( P @ X3 ) @ ( Q2 @ X3 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2505_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I4 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2506_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I4 )
=> ( ( member @ A @ I @ I4 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I4 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2507_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2508_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_2509_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2510_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2511_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_2512_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X3: A] :
( if @ A
@ ( X3
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_2513_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_2514_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T2: set @ B,S: set @ B,H3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H3 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S @ T2 ) )
=> ( ( G2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S @ T2 ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H3 @ T2 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_2515_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ B,A4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B3 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2516_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T2 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2517_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,H3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H3 @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2518_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T2 )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2519_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2520_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A4: set @ B,B3: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C3 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C3 @ A4 ) )
=> ( ( G2 @ A8 )
= ( one_one @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C3 @ B3 ) )
=> ( ( H3 @ B7 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ C3 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ B3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2521_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A4: set @ B,B3: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C3 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C3 @ A4 ) )
=> ( ( G2 @ A8 )
= ( one_one @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C3 @ B3 ) )
=> ( ( H3 @ B7 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ B3 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ C3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2522_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B3 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_2523_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_2524_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( G2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_2525_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B3 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_2526_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_2527_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,H3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H3 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S @ T2 ) )
=> ( ( G2 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S @ T2 ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ T2 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_2528_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] : ( G2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_2529_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( F2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_2530_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_2531_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_2532_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_2533_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_2534_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_2535_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_2536_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_2537_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_2538_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,P: B > $o,H3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_2539_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,P: B > $o,H3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_2540_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_2541_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_2542_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_2543_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less @ A @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2544_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_2545_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
=> ( ( G2 @ X2 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B3 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_2546_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2547_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G2: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2548_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_2549_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,A3: B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2550_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_2551_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B3 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_2552_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2553_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G2: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2554_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B3 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_2555_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B3 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_2556_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ B3 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_2557_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A4: set @ A,B3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_2558_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A4 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ B3 @ A4 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_2559_sum__Un__nat,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_2560_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2561_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: set @ B,F2: B > A,B2: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F2 @ A5 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F2 @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F2
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B2 @ ( F2 @ A5 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_2562_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_2563_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_2564_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2565_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2566_eucl__rel__int__remainderI,axiom,
! [R3: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R3 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_2567_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A4: set @ C,F2: C > B] :
( ( member @ C @ I @ A4 )
=> ( ! [X2: C] :
( ( member @ C @ X2 @ ( minus_minus @ ( set @ C ) @ A4 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X2 ) ) )
=> ( ( finite_finite2 @ C @ A4 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A4 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2568_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B3: set @ A,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B7 ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A8 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2569_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_2570_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A4: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
=> ( ( F2 @ X2 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_2571_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_2572_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2573_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
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q7: int] :
( ( A32
= ( product_Pair @ int @ int @ Q7 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q7 @ A22 ) ) ) )
=> ~ ! [R6: int,Q7: int] :
( ( A32
= ( product_Pair @ int @ int @ Q7 @ R6 ) )
=> ( ( ( sgn_sgn @ int @ R6 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R6 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q7 @ A22 ) @ R6 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_2574_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
( ? [K4: int] :
( ( A12 = K4 )
& ( A23
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K4 ) ) )
| ? [L2: int,K4: int,Q5: int] :
( ( A12 = K4 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K4
= ( times_times @ int @ Q5 @ L2 ) ) )
| ? [R4: int,L2: int,K4: int,Q5: int] :
( ( A12 = K4 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ R4 ) )
& ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ L2 ) )
& ( K4
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R4 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_2575_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_2576_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_2577_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_2578_finite__compl,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_2579_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2580_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2581_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_2582_finite__Collect__subsets,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2583_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ F5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H3 @ F5 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_2584_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q2 ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2585_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_2586_finite__interval__int1,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A3 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2587_finite__interval__int4,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A3 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_2588_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_2589_finite__Int,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G5 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% finite_Int
thf(fact_2590_finite__Un,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) )
= ( ( finite_finite2 @ A @ F5 )
& ( finite_finite2 @ A @ G5 ) ) ) ).
% finite_Un
thf(fact_2591_finite__interval__int2,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A3 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2592_finite__interval__int3,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A3 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2593_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_2594_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_2595_infinite__UNIV__nat,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% infinite_UNIV_nat
thf(fact_2596_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_2597_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B3 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ B3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_2598_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K4: code_integer] :
( if @ code_integer
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_2599_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_2600_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_2601_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_2602_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_2603_ex__new__if__finite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A4 )
=> ? [A8: A] :
~ ( member @ A @ A8 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_2604_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_2605_finite__UnI,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) ) ) ) ).
% finite_UnI
thf(fact_2606_Un__infinite,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_2607_infinite__Un,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_Un
thf(fact_2608_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
=> ( ( ord_less_eq @ A @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_2609_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
=> ( ( ord_less_eq @ A @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_2610_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A10: set @ A] :
( ? [A8: A] :
( A3
= ( insert2 @ A @ A8 @ A10 ) )
=> ~ ( finite_finite2 @ A @ A10 ) ) ) ) ).
% finite.cases
thf(fact_2611_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,B4: A] :
( ( A5
= ( insert2 @ A @ B4 @ A6 ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_2612_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X2 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_2613_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] : ( P @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X2: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X2 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_2614_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ! [A10: set @ A] :
( ~ ( finite_finite2 @ A @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X2 @ F6 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_2615_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_2616_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_2617_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_2618_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A8 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A4 )
=> ( ~ ( member @ A @ A8 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A8 @ F6 ) ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_2619_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A8 @ A4 )
=> ( ~ ( member @ A @ A8 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A8 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_2620_infinite__remove,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_2621_infinite__coinduct,axiom,
! [A: $tType,X7: ( set @ A ) > $o,A4: set @ A] :
( ( X7 @ A4 )
=> ( ! [A10: set @ A] :
( ( X7 @ A10 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A10 )
& ( ( X7 @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_2622_finite__empty__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ A4 )
=> ( ! [A8: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( member @ A @ A8 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_2623_finite__remove__induct,axiom,
! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_2624_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B3 )
=> ( P @ B3 ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_2625_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T6: set @ A] :
( ( ord_less @ ( set @ A ) @ T6 @ S )
=> ( ( P @ T6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( minus_minus @ ( set @ A ) @ S @ T6 ) )
& ( P @ ( insert2 @ A @ X5 @ T6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_2626_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_2627_diff__preserves__multiset,axiom,
! [A: $tType,M4: A > nat,N4: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M4 @ X3 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_2628_add__mset__in__multiset,axiom,
! [A: $tType,M4: A > nat,A3: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M4 @ X3 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X3 = A3 ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_2629_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B7: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A10 )
=> ( ord_less @ A @ B7 @ X5 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert2 @ A @ B7 @ A10 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_2630_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B7: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A10 )
=> ( ord_less @ A @ X5 @ B7 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert2 @ A @ B7 @ A10 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_2631_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y6: B] :
( ( member @ B @ Y6 @ S6 )
=> ( ord_less_eq @ A @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert2 @ B @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_2632_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X7: set @ A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X7 )
& ( ord_less @ A @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X7 ) ) ) ) ).
% infinite_growing
thf(fact_2633_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ S )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S )
& ( ord_less @ A @ Xa2 @ X2 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_2634_filter__preserves__multiset,axiom,
! [A: $tType,M4: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M4 @ X3 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_2635_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K4: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K4 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K4 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_2636_rec__nat__Suc__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N: nat] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( suc @ N ) )
= ( F22 @ N @ ( F2 @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_2637_rec__nat__0__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_2638_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I4: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I4 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I4 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I4 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I4 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_2639_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A4 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_2640_nat__not__finite,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% nat_not_finite
thf(fact_2641_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_2642_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_2643_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ( finite_card @ A @ S )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_2644_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_2645_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_2646_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_2647_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_2648_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_2649_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_2650_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : Y
@ A4 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A4 ) ) ) ) ).
% prod_constant
thf(fact_2651_card__0__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_2652_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_2653_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_2654_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_2655_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : Y
@ A4 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_2656_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ A3 @ B3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B3 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_2657_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_2658_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( P4 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert2 @ B @ I @ I4 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I4 ) ) )
& ( ~ ( member @ B @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert2 @ B @ I @ I4 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I4 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_2659_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( finite_card @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_2660_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_2661_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( zero_neq_one_of_bool @ A @ ( P @ X3 ) )
@ A4 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_2662_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_2663_n__subsets,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
& ( ( finite_card @ A @ B5 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A4 ) @ K ) ) ) ).
% n_subsets
thf(fact_2664_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,I4: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G2 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I4 ) ) ) ).
% sum.non_neutral'
thf(fact_2665_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( A4
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_2666_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_2667_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S3: set @ A,T4: set @ B,R: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T4 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_2668_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_2669_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( ( finite_card @ A @ A6 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_2670_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G2 @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_2671_card__eq__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_2672_card__1__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( one_one @ nat ) )
=> ~ ! [X2: A] :
( A4
!= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_2673_card__Un__le,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B3 ) ) ) ).
% card_Un_le
thf(fact_2674_card__less,axiom,
! [M4: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M4 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_2675_card__less__Suc,axiom,
! [M4: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ ( suc @ K4 ) @ M4 )
& ( ord_less @ nat @ K4 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M4 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_2676_card__less__Suc2,axiom,
! [M4: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ ( suc @ K4 ) @ M4 )
& ( ord_less @ nat @ K4 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M4 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_2677_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_2678_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_2679_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_2680_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A4: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( suc @ ( F2 @ X3 ) )
@ A4 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( finite_card @ A @ A4 ) ) ) ).
% sum_Suc
thf(fact_2681_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S )
& ( R @ I3 @ X2 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R @ I3 @ J3 ) ) ) )
@ S )
= ( times_times @ nat @ K @ ( finite_card @ B @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_2682_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_2683_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_2684_card__gt__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_2685_card__Suc__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B4: A,B5: set @ A] :
( ( A4
= ( insert2 @ A @ B4 @ B5 ) )
& ~ ( member @ A @ B4 @ B5 )
& ( ( finite_card @ A @ B5 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_2686_card__eq__SucD,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
=> ? [B7: A,B10: set @ A] :
( ( A4
= ( insert2 @ A @ B7 @ B10 ) )
& ~ ( member @ A @ B7 @ B10 )
& ( ( finite_card @ A @ B10 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B10
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_2687_card__1__singleton__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X3: A] :
( A4
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_2688_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,K5: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ K5 @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_bounded_below
thf(fact_2689_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_2690_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_2691_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S )
=> ( S
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_2692_card__Diff1__le,axiom,
! [A: $tType,A4: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ).
% card_Diff1_le
thf(fact_2693_card__Un__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B3 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_2694_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G2 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( H3 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G2 @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_2695_card__Diff__subset__Int,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_2696_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_2697_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I5: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I5 )
& ( ( P6 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I5 )
& ( ( P6 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_2698_card__sum__le__nat__sum,axiom,
! [S: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_2699_subset__Collect__iff,axiom,
! [A: $tType,B3: set @ A,A4: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_2700_subset__CollectI,axiom,
! [A: $tType,B3: set @ A,A4: set @ A,Q2: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B3 )
=> ( ( Q2 @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B3 )
& ( Q2 @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_2701_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X3: A,Y3: A] :
( ( S
= ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X3 != Y3 ) ) ) ) ).
% card_2_iff
thf(fact_2702_card_Oremove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_2703_card_Oinsert__remove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_2704_card__Suc__Diff1,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_2705_card__Diff1__less__iff,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) )
= ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_2706_card__Diff2__less,axiom,
! [A: $tType,A4: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_2707_card__Diff1__less,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_2708_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X3: A,Y3: A,Z5: A] :
( ( S
= ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X3 != Y3 )
& ( Y3 != Z5 )
& ( X3 != Z5 ) ) ) ) ).
% card_3_iff
thf(fact_2709_odd__card__imp__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) )
=> ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_2710_card__Un__disjoint,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_2711_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_2712_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_2713_card__Diff__singleton,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_2714_nat__mult__distrib__neg,axiom,
! [Z2: int,Z9: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z9 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z9 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_2715_diff__nat__eq__if,axiom,
! [Z9: int,Z2: int] :
( ( ( ord_less @ int @ Z9 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z9 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less @ int @ Z9 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z9 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z9 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z9 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_2716_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,N: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_2717_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_2718_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less @ A @ ( F2 @ I2 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A4 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_2719_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K4: int] : ( if @ A @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K4 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K4 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_2720_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_2721_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
@ S )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
@ S )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_2722_finite__transitivity__chain,axiom,
! [A: $tType,A4: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X2: A] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: A,Y2: A,Z3: A] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A4 )
& ( R @ X2 @ Y6 ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_2723_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I4: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I4 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I4 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I4 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_2724_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_2725_card__UNIV__char,axiom,
( ( finite_card @ char @ ( top_top @ ( set @ char ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_2726_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_2727_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_2728_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_2729_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( 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 ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K4 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K4 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S2 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_2730_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_2731_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_2732_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_2733_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(8)
thf(fact_2734_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_2735_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_2736_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_2737_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_2738_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_2739_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_2740_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(5)
thf(fact_2741_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_2742_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_2743_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C2: B,F2: A > C] :
( ( comp @ C @ B @ A
@ ^ [X3: C] : C2
@ F2 )
= ( ^ [X3: A] : C2 ) ) ).
% K_record_comp
thf(fact_2744_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert2 @ $o @ $false @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_2745_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F4: $tType,Rep: A > B,Abs: B > A,Rep3: C > D,Abs3: D > C,Rep4: E > F4,Abs4: F4 > E,M4: F4 > D,M1: B > D,M22: F4 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep3 @ Abs3 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F4 @ Rep4 @ Abs4 @ ( top_top @ ( set @ F4 ) ) )
=> ( ( ( comp @ F4 @ C @ E @ ( comp @ D @ C @ F4 @ Abs3 @ M4 ) @ Rep4 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs3 @ M1 ) @ Rep ) @ ( comp @ F4 @ A @ E @ ( comp @ B @ A @ F4 @ Abs @ M22 ) @ Rep4 ) ) )
=> ( ( comp @ B @ D @ F4 @ M1 @ M22 )
= M4 ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_2746_type__copy__map__comp0,axiom,
! [F4: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M4: C > D,M1: B > D,M22: C > B,F2: D > F4,G2: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M4
= ( comp @ B @ D @ C @ M1 @ M22 ) )
=> ( ( comp @ C @ F4 @ E @ ( comp @ D @ F4 @ C @ F2 @ M4 ) @ G2 )
= ( comp @ A @ F4 @ E @ ( comp @ B @ F4 @ A @ ( comp @ D @ F4 @ B @ F2 @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M22 ) @ G2 ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_2747_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.comm_monoid_axioms
thf(fact_2748_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_2749_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.semilattice_neutr_axioms
thf(fact_2750_set__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N2: nat,K4: int] : ( bit_se1065995026697491101ons_or @ int @ K4 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).
% set_bit_int_def
thf(fact_2751_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M2: nat,N2: nat] : ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_2752_or__not__numerals_I1_J,axiom,
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(1)
thf(fact_2753_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_2754_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_2755_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_2756_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_2757_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_2758_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_2759_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_2760_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 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_2761_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_2762_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_2763_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% mask_Suc_double
thf(fact_2764_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_2765_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_2766_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_2767_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 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_2768_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K4: int,L2: int] :
( if @ int
@ ( ( K4
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K4
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K4
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_2769_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_2770_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_2771_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_2772_card__partition,axiom,
! [A: $tType,C3: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C3 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ( ! [C4: set @ A] :
( ( member @ ( set @ A ) @ C4 @ C3 )
=> ( ( finite_card @ A @ C4 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C3 )
=> ( ( member @ ( set @ A ) @ C22 @ C3 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C3 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_2773_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.right_idem
thf(fact_2774_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.left_idem
thf(fact_2775_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_2776_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_2777_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_2778_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_2779_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% max_less_iff_conj
thf(fact_2780_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_2781_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_2782_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_2783_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_2784_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_2785_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_2786_max__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(1)
thf(fact_2787_max__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(2)
thf(fact_2788_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_2789_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_2790_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_2791_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_2792_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_2793_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_2794_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_2795_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_2796_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_2797_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_2798_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_2799_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_2800_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A4: C] :
( ( member @ A @ X @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A4 ) )
=> ( member @ A @ X @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( comp @ B @ ( set @ ( set @ A ) ) @ C @ Gset @ Gmap ) @ A4 ) ) ) ).
% in_Union_o_assoc
thf(fact_2801_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def
thf(fact_2802_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_2803_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_2804_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D3 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_2805_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_2806_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_max @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% max.orderI
thf(fact_2807_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_2808_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_2809_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B4 ) ) ) ) ) ).
% max.order_iff
thf(fact_2810_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_2811_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_2812_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_2813_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_max @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_2814_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_max @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% max.absorb_iff2
thf(fact_2815_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_2816_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_2817_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z2 @ X )
| ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_2818_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_2819_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_2820_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_2821_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_2822_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_2823_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C2 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_2824_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( ord_max @ A @ B4 @ A5 ) ) ) ) ).
% max.commute
thf(fact_2825_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C2 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_2826_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,Y: A,Z2: A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Conj @ ( Xor @ Y @ Z2 ) @ X )
= ( Xor @ ( Conj @ Y @ X ) @ ( Conj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
thf(fact_2827_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Compl @ X ) )
= One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
thf(fact_2828_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A,Z2: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Conj @ X @ ( Xor @ Y @ Z2 ) )
= ( Xor @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
thf(fact_2829_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Compl @ Y ) )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
thf(fact_2830_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ X )
= One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
thf(fact_2831_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ Y )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
thf(fact_2832_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ One )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
thf(fact_2833_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Xor @ X @ Y ) )
= Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
thf(fact_2834_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ One @ X )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
thf(fact_2835_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ X )
= Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
thf(fact_2836_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ Y )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
thf(fact_2837_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ Y )
= ( Disj @ ( Conj @ X @ ( Compl @ Y ) ) @ ( Conj @ ( Compl @ X ) @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
thf(fact_2838_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_2839_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_2840_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_2841_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_2842_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ ( set @ B ),G2: B > A] :
( ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ B3 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [A13: set @ B] :
( ( member @ ( set @ B ) @ A13 @ B3 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B3 )
=> ( ( A13 != A24 )
=> ! [X2: B] :
( ( member @ B @ X2 @ A13 )
=> ( ( member @ B @ X2 @ A24 )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G2 @ B3 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_2843_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_2844_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ ( set @ B ),G2: B > A] :
( ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ C3 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ C3 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C3 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G2 @ C3 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_2845_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ ( set @ B ),G2: B > A] :
( ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ C3 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ C3 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C3 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G2 @ C3 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_2846_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > 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)
thf(fact_2847_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B3 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( ( inf_inf @ A @ X3 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_2848_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X: C,S: B > ( set @ C ),Y: B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X @ ( comp @ B @ ( set @ C ) @ A @ S @ Rep @ ( Abs @ Y ) ) )
=> ( member @ C @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_2849_insert__partition,axiom,
! [A: $tType,X: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F5 )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ ( insert2 @ ( set @ A ) @ X @ F5 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert2 @ ( set @ A ) @ X @ F5 ) )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_2850_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( member @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ A4 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_2851_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_2852_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_2853_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_2854_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(1)
thf(fact_2855_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_2856_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ L ) ) @ E4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E4 ) ) ) ) ).
% cSup_asclose
thf(fact_2857_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_2858_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_2859_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_2860_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_2861_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_2862_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_2863_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_2864_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_2865_dvd__partition,axiom,
! [A: $tType,C3: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X2 ) ) )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C3 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_2866_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_2867_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_2868_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_2869_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_2870_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_2871_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_2872_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_insert
thf(fact_2873_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_2874_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_2875_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_2876_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_2877_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( ord_less @ A @ X3 @ Y3 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_2878_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( X3
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_2879_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( X3
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_2880_Union__Un__distrib,axiom,
! [A: $tType,A4: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_Un_distrib
thf(fact_2881_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_2882_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_2883_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.comm_monoid_axioms
thf(fact_2884_max__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.monoid_axioms
thf(fact_2885_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ M5 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_2886_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ N @ M5 ) )
@ M ) ) ).
% max_Suc1
thf(fact_2887_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq @ nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less @ nat @ Y3 @ X3 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_2888_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_2889_empty__Union__conv,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( X3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_2890_Union__empty__conv,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( X3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_2891_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_2892_Union__insert,axiom,
! [A: $tType,A3: set @ A,B3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_insert
thf(fact_2893_complete__linorder__sup__max,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% complete_linorder_sup_max
thf(fact_2894_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A4 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_2895_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,Z2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X7 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_2896_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A3: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ( ord_less_eq @ A @ X2 @ A3 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X7 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_2897_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Z2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X7 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ X7 )
& ( ord_less @ A @ Z2 @ X2 ) ) ) ) ) ).
% less_cSupD
thf(fact_2898_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X7: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X7 ) )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ) ) ).
% less_cSupE
thf(fact_2899_Sup__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_union_distrib
thf(fact_2900_Union__disjoint,axiom,
! [A: $tType,C3: set @ ( set @ A ),A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_2901_Union__Int__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A4 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_Int_subset
thf(fact_2902_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_2903_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,A3: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X7 ) @ A3 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ( ord_less @ A @ X3 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_2904_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_2905_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_2906_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_2907_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_2908_card__UNION,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A4 )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( finite_finite2 @ A @ X2 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I5: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I5 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I5 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I5 @ A4 )
& ( I5
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_2909_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_finite_insert
thf(fact_2910_top__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( top_top @ A )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_finite_def
thf(fact_2911_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S: set @ A] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I2 ) @ S )
=> ( ( finite_finite2 @ A @ S )
=> ( ? [N5: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N5 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N5 )
=> ( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N5 @ N3 )
=> ( ( F2 @ N5 )
= ( F2 @ N3 ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_2912_image__ident,axiom,
! [A: $tType,Y4: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : X3
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_2913_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F2 @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_2914_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F2 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_2915_image__empty,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_2916_Inf__top__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( X3
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_2917_Inf__top__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( X3
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_2918_Inter__UNIV__conv_I1_J,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ A4 )
= ( top_top @ ( set @ A ) ) )
= ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( X3
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_2919_Inter__UNIV__conv_I2_J,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ A4 ) )
= ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( X3
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_2920_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F2: C > B > A,A4: set @ C,X: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F2 @ A4 ) @ X )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y3: C] : ( F2 @ Y3 @ X )
@ A4 ) ) ) ) ).
% SUP_apply
thf(fact_2921_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A4: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ).
% SUP_identity_eq
thf(fact_2922_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F2: C > B > A,A4: set @ C,X: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F2 @ A4 ) @ X )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [Y3: C] : ( F2 @ Y3 @ X )
@ A4 ) ) ) ) ).
% INF_apply
thf(fact_2923_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A4: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ).
% INF_identity_eq
thf(fact_2924_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( member @ A @ B2 @ ( B3 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_2925_UN__I,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B3: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ B @ B2 @ ( B3 @ A3 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_2926_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( member @ A @ B2 @ ( B3 @ X3 ) ) ) ) ) ).
% INT_iff
thf(fact_2927_INT__I,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: B,B3: A > ( set @ B )] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ B @ B2 @ ( B3 @ X2 ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ).
% INT_I
thf(fact_2928_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_2929_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_2930_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_2931_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_2932_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_2933_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_insert
thf(fact_2934_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_2935_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_2936_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A4: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B3 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_2937_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B3 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_2938_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( bot_bot @ A )
@ A4 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_2939_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_2940_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F2
@ A4 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_2941_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_2942_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F2
@ A4 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_2943_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_2944_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_2945_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( top_top @ A )
@ A4 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_2946_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B3 @ A4 ) )
= ( top_top @ A ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_2947_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A4: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B3 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_2948_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_2949_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: B > A,G2: B > A,S: set @ B] :
( ( image2 @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ S )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ S @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G2
@ ( inf_inf @ ( set @ B ) @ S
@ ( collect @ B
@ ^ [X3: B] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2950_UN__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_2951_finite__UN__I,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( finite_finite2 @ B @ ( B3 @ A8 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_2952_UN__Un,axiom,
! [A: $tType,B: $tType,M4: B > ( set @ A ),A4: set @ B,B3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M4 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M4 @ A4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M4 @ B3 ) ) ) ) ).
% UN_Un
thf(fact_2953_finite__INT,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B )] :
( ? [X5: A] :
( ( member @ A @ X5 @ I4 )
& ( finite_finite2 @ B @ ( A4 @ X5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) ) ) ).
% finite_INT
thf(fact_2954_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ A] :
( ( image2 @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A4 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_2955_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A4 )
& ( ord_less @ A @ ( F2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_2956_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( top_top @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A4 )
& ( ord_less @ A @ X3 @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_2957_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_2958_UN__singleton,axiom,
! [A: $tType,A4: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_2959_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ B,A3: A,B3: B > ( set @ A )] :
( ( ( C3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ A3 @ ( B3 @ X3 ) )
@ C3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ A3 @ ( B3 @ X3 ) )
@ C3 ) )
= ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2960_UN__simps_I3_J,axiom,
! [E: $tType,F4: $tType,C3: set @ F4,A4: set @ E,B3: F4 > ( set @ E )] :
( ( ( C3
= ( bot_bot @ ( set @ F4 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F4 @ ( set @ E )
@ ^ [X3: F4] : ( sup_sup @ ( set @ E ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ F4 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F4 @ ( set @ E )
@ ^ [X3: F4] : ( sup_sup @ ( set @ E ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) )
= ( sup_sup @ ( set @ E ) @ A4 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F4 @ ( set @ E ) @ B3 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2961_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C3: set @ C,A4: C > ( set @ D ),B3: set @ D] :
( ( ( C3
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X3: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X3: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C3 ) ) @ B3 ) ) ) ) ).
% UN_simps(2)
thf(fact_2962_UN__insert,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: B,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B3 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2963_INT__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A4 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_2964_INT__insert,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: B,A4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B3 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2965_Compl__INT,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( uminus_uminus @ ( set @ A ) @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% Compl_INT
thf(fact_2966_Compl__UN,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( uminus_uminus @ ( set @ A ) @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% Compl_UN
thf(fact_2967_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_2968_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_2969_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C5: A] : ( divide_divide @ A @ C5 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_2970_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ A,A4: A > ( set @ B ),B3: set @ B] :
( ( ( C3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C3 ) ) @ B3 ) ) ) ) ).
% INT_simps(1)
thf(fact_2971_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C3: set @ D,A4: set @ C,B3: D > ( set @ C )] :
( ( ( C3
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X3: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X3: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) )
= ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B3 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2972_INT__simps_I3_J,axiom,
! [E: $tType,F4: $tType,C3: set @ E,A4: E > ( set @ F4 ),B3: set @ F4] :
( ( ( C3
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F4 )
@ ( image2 @ E @ ( set @ F4 )
@ ^ [X3: E] : ( minus_minus @ ( set @ F4 ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) )
= ( top_top @ ( set @ F4 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F4 )
@ ( image2 @ E @ ( set @ F4 )
@ ^ [X3: E] : ( minus_minus @ ( set @ F4 ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) )
= ( minus_minus @ ( set @ F4 ) @ ( complete_Inf_Inf @ ( set @ F4 ) @ ( image2 @ E @ ( set @ F4 ) @ A4 @ C3 ) ) @ B3 ) ) ) ) ).
% INT_simps(3)
thf(fact_2973_INT__simps_I4_J,axiom,
! [G3: $tType,H8: $tType,C3: set @ H8,A4: set @ G3,B3: H8 > ( set @ G3 )] :
( ( ( C3
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X3: H8] : ( minus_minus @ ( set @ G3 ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) )
= ( top_top @ ( set @ G3 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X3: H8] : ( minus_minus @ ( set @ G3 ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) )
= ( minus_minus @ ( set @ G3 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B3 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2974_Union__natural,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ ( set @ ( set @ B ) ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F2 ) ) )
= ( comp @ ( set @ A ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( image2 @ A @ B @ F2 ) @ ( complete_Sup_Sup @ ( set @ A ) ) ) ) ).
% Union_natural
thf(fact_2975_uminus__Inf,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A4: set @ A] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ A4 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ) ) ).
% uminus_Inf
thf(fact_2976_uminus__Sup,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A4: set @ A] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ A4 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ) ) ).
% uminus_Sup
thf(fact_2977_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_2978_UN__extend__simps_I10_J,axiom,
! [V4: $tType,U3: $tType,T: $tType,B3: U3 > ( set @ V4 ),F2: T > U3,A4: set @ T] :
( ( complete_Sup_Sup @ ( set @ V4 )
@ ( image2 @ T @ ( set @ V4 )
@ ^ [A5: T] : ( B3 @ ( F2 @ A5 ) )
@ A4 ) )
= ( complete_Sup_Sup @ ( set @ V4 ) @ ( image2 @ U3 @ ( set @ V4 ) @ B3 @ ( image2 @ T @ U3 @ F2 @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_2979_UN__extend__simps_I8_J,axiom,
! [P8: $tType,O2: $tType,B3: O2 > ( set @ P8 ),A4: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P8 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P8 )
@ ^ [Y3: set @ O2] : ( complete_Sup_Sup @ ( set @ P8 ) @ ( image2 @ O2 @ ( set @ P8 ) @ B3 @ Y3 ) )
@ A4 ) )
= ( complete_Sup_Sup @ ( set @ P8 ) @ ( image2 @ O2 @ ( set @ P8 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A4 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_2980_INT__extend__simps_I9_J,axiom,
! [S7: $tType,R7: $tType,Q8: $tType,C3: R7 > ( set @ S7 ),B3: Q8 > ( set @ R7 ),A4: set @ Q8] :
( ( complete_Inf_Inf @ ( set @ S7 )
@ ( image2 @ Q8 @ ( set @ S7 )
@ ^ [X3: Q8] : ( complete_Inf_Inf @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C3 @ ( B3 @ X3 ) ) )
@ A4 ) )
= ( complete_Inf_Inf @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image2 @ Q8 @ ( set @ R7 ) @ B3 @ A4 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_2981_INT__extend__simps_I8_J,axiom,
! [P8: $tType,O2: $tType,B3: O2 > ( set @ P8 ),A4: set @ ( set @ O2 )] :
( ( complete_Inf_Inf @ ( set @ P8 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P8 )
@ ^ [Y3: set @ O2] : ( complete_Inf_Inf @ ( set @ P8 ) @ ( image2 @ O2 @ ( set @ P8 ) @ B3 @ Y3 ) )
@ A4 ) )
= ( complete_Inf_Inf @ ( set @ P8 ) @ ( image2 @ O2 @ ( set @ P8 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A4 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_2982_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,B3: C > ( set @ B ),A4: set @ C] :
( ( image2 @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( image2 @ B @ A @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% image_UN
thf(fact_2983_image__Union,axiom,
! [A: $tType,B: $tType,F2: B > A,S: set @ ( set @ B )] :
( ( image2 @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F2 ) @ S ) ) ) ).
% image_Union
thf(fact_2984_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,M: A,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X2 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_2985_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,C2: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= C2 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( ( F2 @ X3 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2986_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,X: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( ( F2 @ I2 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_2987_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A4: set @ A,A3: B,B3: A > ( set @ B )] :
( ( member @ A @ U @ A4 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( insert2 @ B @ A3 @ ( B3 @ X3 ) )
@ A4 ) )
= ( insert2 @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2988_INT__extend__simps_I5_J,axiom,
! [I6: $tType,J4: $tType,A3: I6,B3: J4 > ( set @ I6 ),C3: set @ J4] :
( ( insert2 @ I6 @ A3 @ ( complete_Inf_Inf @ ( set @ I6 ) @ ( image2 @ J4 @ ( set @ I6 ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ I6 )
@ ( image2 @ J4 @ ( set @ I6 )
@ ^ [X3: J4] : ( insert2 @ I6 @ A3 @ ( B3 @ X3 ) )
@ C3 ) ) ) ).
% INT_extend_simps(5)
thf(fact_2989_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ B,A4: set @ B,F2: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B3 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2990_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B3: set @ A,A4: B > ( set @ A ),I4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( A4 @ X3 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_2991_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ A,F2: A > ( set @ B ),G2: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G2 @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_2992_INT__greatest,axiom,
! [B: $tType,A: $tType,A4: set @ A,C3: set @ B,B3: A > ( set @ B )] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ C3 @ ( B3 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_2993_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) @ ( B3 @ A3 ) ) ) ).
% INT_lower
thf(fact_2994_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_greatest
thf(fact_2995_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X3 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_2996_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A,U: A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_2997_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G2: B > A,A4: set @ B] :
( ! [X2: B] : ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% INF_mono'
thf(fact_2998_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( F2 @ I ) ) ) ) ).
% INF_lower
thf(fact_2999_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ B,A4: set @ C,F2: C > A,G2: B > A] :
( ! [M3: B] :
( ( member @ B @ M3 @ B3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G2 @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B3 ) ) ) ) ) ).
% INF_mono
thf(fact_3000_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,X: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ X @ ( F2 @ I2 ) ) )
=> ( ! [Y2: A] :
( ! [I7: B] :
( ( member @ B @ I7 @ A4 )
=> ( ord_less_eq @ A @ Y2 @ ( F2 @ I7 ) ) )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_3001_vimage__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,B3: C > ( set @ B ),A4: set @ C] :
( ( vimage @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( vimage @ A @ B @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% vimage_INT
thf(fact_3002_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B,B3: C > ( set @ A ),J5: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B3 @ J5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( sup_sup @ ( set @ A ) @ ( A4 @ I3 ) @ ( B3 @ J3 ) )
@ J5 ) )
@ I4 ) ) ) ).
% Un_INT_distrib2
thf(fact_3003_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B3: set @ A,A4: B > ( set @ A ),I4: set @ B] :
( ( sup_sup @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( sup_sup @ ( set @ A ) @ B3 @ ( A4 @ I3 ) )
@ I4 ) ) ) ).
% Un_INT_distrib
thf(fact_3004_INT__extend__simps_I6_J,axiom,
! [L5: $tType,K6: $tType,A4: K6 > ( set @ L5 ),C3: set @ K6,B3: set @ L5] :
( ( sup_sup @ ( set @ L5 ) @ ( complete_Inf_Inf @ ( set @ L5 ) @ ( image2 @ K6 @ ( set @ L5 ) @ A4 @ C3 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ L5 )
@ ( image2 @ K6 @ ( set @ L5 )
@ ^ [X3: K6] : ( sup_sup @ ( set @ L5 ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) ) ) ).
% INT_extend_simps(6)
thf(fact_3005_INT__extend__simps_I7_J,axiom,
! [M6: $tType,N6: $tType,A4: set @ M6,B3: N6 > ( set @ M6 ),C3: set @ N6] :
( ( sup_sup @ ( set @ M6 ) @ A4 @ ( complete_Inf_Inf @ ( set @ M6 ) @ ( image2 @ N6 @ ( set @ M6 ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ M6 )
@ ( image2 @ N6 @ ( set @ M6 )
@ ^ [X3: N6] : ( sup_sup @ ( set @ M6 ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) ) ) ).
% INT_extend_simps(7)
thf(fact_3006_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),B3: B > ( set @ A ),C3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( inf_inf @ ( set @ A ) @ ( A4 @ X3 ) @ ( B3 @ X3 ) )
@ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ C3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Int_Inter_image
thf(fact_3007_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),B3: B > ( set @ A ),I4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( inf_inf @ ( set @ A ) @ ( A4 @ I3 ) @ ( B3 @ I3 ) )
@ I4 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ I4 ) ) ) ) ).
% INT_Int_distrib
thf(fact_3008_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I4: set @ A,A4: A > ( set @ B )] :
( ( member @ A @ K @ I4 )
=> ( ( inf_inf @ ( set @ B ) @ ( A4 @ K ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) )
= ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) ) ) ).
% INT_absorb
thf(fact_3009_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I4: set @ B,A4: B > A] :
( ( member @ B @ K @ I4 )
=> ( ( inf_inf @ A @ ( A4 @ K ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A4 @ I4 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A4 @ I4 ) ) ) ) ) ).
% INF_absorb
thf(fact_3010_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,G2: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ).
% INF_inf_distrib
thf(fact_3011_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B3 @ A3 ) )
=> ~ ( member @ B @ A3 @ A4 ) ) ) ).
% INT_E
thf(fact_3012_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
=> ( ( member @ B @ A3 @ A4 )
=> ( member @ A @ B2 @ ( B3 @ A3 ) ) ) ) ).
% INT_D
thf(fact_3013_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > C > A,B3: set @ C,A4: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( F2 @ I3 ) @ B3 ) )
@ A4 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( F2 @ I3 @ J3 )
@ A4 ) )
@ B3 ) ) ) ) ).
% INF_commute
thf(fact_3014_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
( ( Sup
@ ( image2 @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_identity_eq
thf(fact_3015_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
( ( Inf
@ ( image2 @ A @ A
@ ^ [X3: A] : X3
@ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_identity_eq
thf(fact_3016_INT__extend__simps_I10_J,axiom,
! [V4: $tType,U3: $tType,T: $tType,B3: U3 > ( set @ V4 ),F2: T > U3,A4: set @ T] :
( ( complete_Inf_Inf @ ( set @ V4 )
@ ( image2 @ T @ ( set @ V4 )
@ ^ [A5: T] : ( B3 @ ( F2 @ A5 ) )
@ A4 ) )
= ( complete_Inf_Inf @ ( set @ V4 ) @ ( image2 @ U3 @ ( set @ V4 ) @ B3 @ ( image2 @ T @ U3 @ F2 @ A4 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_3017_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( ord_less @ A @ ( F2 @ X3 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_3018_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F2: B > A,A4: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_3019_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_3020_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_3021_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A4: set @ B,G2: C > A,B3: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G2 @ B3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B4: C] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G2 @ B4 ) )
@ B3 ) )
@ A4 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_3022_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F2: B > A,B3: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B4: B] : ( sup_sup @ A @ A3 @ ( F2 @ B4 ) )
@ B3 ) ) ) ) ).
% sup_INF
thf(fact_3023_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B3 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B4: A] : ( sup_sup @ A @ B4 @ A3 )
@ B3 ) ) ) ) ).
% Inf_sup
thf(fact_3024_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B3: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B3 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B4: B] : ( sup_sup @ A @ ( F2 @ B4 ) @ A3 )
@ B3 ) ) ) ) ).
% INF_sup
thf(fact_3025_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B3: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B3 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A3 ) @ B3 ) ) ) ) ).
% sup_Inf
thf(fact_3026_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ( P @ X3 ) ) )
= ( image2 @ B @ A @ F2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_3027_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G2: C > B,A4: set @ C] :
( ( image2 @ B @ A @ F2 @ ( image2 @ C @ B @ G2 @ A4 ) )
= ( image2 @ C @ A
@ ^ [X3: C] : ( F2 @ ( G2 @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_3028_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,A4: set @ B] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ~ ! [X2: B] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member @ B @ X2 @ A4 ) ) ) ).
% imageE
thf(fact_3029_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A4 ) )
= ( top_top @ A ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_3030_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_3031_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,X: A,F2: B > A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ X @ ( F2 @ I3 ) )
@ I4 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3032_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,X: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ ( F2 @ I3 ) @ X )
@ I4 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_3033_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B3: B > A,A4: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( uminus_uminus @ A @ ( B3 @ X3 ) )
@ A4 ) ) ) ) ).
% uminus_SUP
thf(fact_3034_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B3: B > A,A4: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( uminus_uminus @ A @ ( B3 @ X3 ) )
@ A4 ) ) ) ) ).
% uminus_INF
thf(fact_3035_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: B,A4: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ A @ ( F2 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_insert
thf(fact_3036_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X: B] : ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_3037_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X: B] :
( ( B2
= ( F2 @ X ) )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_3038_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M4: B > A,A4: set @ B,B3: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M4 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M4 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M4 @ B3 ) ) ) ) ) ).
% INF_union
thf(fact_3039_INT__empty,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_3040_Inter__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B3: set @ A] :
( ! [X8: set @ A] :
( ( member @ ( set @ A ) @ X8 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ B3 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ B3 ) ) ) ).
% Inter_subset
thf(fact_3041_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C3: set @ A,A4: A > ( set @ B ),B3: set @ B] :
( ( ( C3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C3 ) ) @ B3 )
= B3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C3 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3042_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C3: set @ D,A4: set @ C,B3: D > ( set @ C )] :
( ( ( C3
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B3 @ C3 ) ) )
= A4 ) )
& ( ( C3
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X3: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3043_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G2: C > ( set @ B ),A4: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ G2 @ A4 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y3: C] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( G2 @ Y3 ) ) )
@ A4 ) ) ) ) ).
% SUP_UNION
thf(fact_3044_image__Un,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B3: set @ B] :
( ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ ( image2 @ B @ A @ F2 @ B3 ) ) ) ).
% image_Un
thf(fact_3045_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_3046_INT__Un,axiom,
! [A: $tType,B: $tType,M4: B > ( set @ A ),A4: set @ B,B3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M4 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M4 @ A4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M4 @ B3 ) ) ) ) ).
% INT_Un
thf(fact_3047_Inter__Un__distrib,axiom,
! [A: $tType,A4: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) ) ).
% Inter_Un_distrib
thf(fact_3048_UN__extend__simps_I7_J,axiom,
! [M6: $tType,N6: $tType,A4: set @ M6,B3: N6 > ( set @ M6 ),C3: set @ N6] :
( ( minus_minus @ ( set @ M6 ) @ A4 @ ( complete_Inf_Inf @ ( set @ M6 ) @ ( image2 @ N6 @ ( set @ M6 ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ M6 )
@ ( image2 @ N6 @ ( set @ M6 )
@ ^ [X3: N6] : ( minus_minus @ ( set @ M6 ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) ) ) ).
% UN_extend_simps(7)
thf(fact_3049_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [X3: C] : ( P @ X3 @ Y3 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [F: B > C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( P @ ( F @ X3 ) @ X3 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_3050_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [X3: C] : ( P @ X3 @ Y3 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [X3: B > C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : ( P @ ( X3 @ Y3 ) @ Y3 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_3051_bot__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( bot_bot @ A )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bot_finite_def
thf(fact_3052_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A4: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_finite_insert
thf(fact_3053_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_3054_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( ( F2 @ A5 )
= ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3055_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B3: set @ B] :
( ! [X2: A] :
( ( P @ X2 )
=> ( member @ B @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_3056_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G2: B > C] :
( ( image2 @ B @ A
@ ^ [X3: B] : ( F2 @ ( G2 @ X3 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F2 @ ( image2 @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_3057_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X2: B] :
( B2
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_3058_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > C > A,B3: set @ C,A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( F2 @ I3 ) @ B3 ) )
@ A4 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( F2 @ I3 @ J3 )
@ A4 ) )
@ B3 ) ) ) ) ).
% SUP_commute
thf(fact_3059_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C3: B > ( set @ A ),B3: C > ( set @ B ),A4: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [Y3: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C3 @ ( B3 @ Y3 ) ) )
@ A4 ) ) ) ).
% UN_UN_flatten
thf(fact_3060_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
=> ~ ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ~ ( member @ A @ B2 @ ( B3 @ X2 ) ) ) ) ).
% UN_E
thf(fact_3061_UN__extend__simps_I9_J,axiom,
! [S7: $tType,R7: $tType,Q8: $tType,C3: R7 > ( set @ S7 ),B3: Q8 > ( set @ R7 ),A4: set @ Q8] :
( ( complete_Sup_Sup @ ( set @ S7 )
@ ( image2 @ Q8 @ ( set @ S7 )
@ ^ [X3: Q8] : ( complete_Sup_Sup @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C3 @ ( B3 @ X3 ) ) )
@ A4 ) )
= ( complete_Sup_Sup @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image2 @ Q8 @ ( set @ R7 ) @ B3 @ A4 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_3062_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ ( F2 @ X3 ) @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) )
= ( image2 @ B @ A @ F2 @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_3063_INT__extend__simps_I3_J,axiom,
! [F4: $tType,E: $tType,C3: set @ E,A4: E > ( set @ F4 ),B3: set @ F4] :
( ( ( C3
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F4 ) @ ( complete_Inf_Inf @ ( set @ F4 ) @ ( image2 @ E @ ( set @ F4 ) @ A4 @ C3 ) ) @ B3 )
= ( minus_minus @ ( set @ F4 ) @ ( top_top @ ( set @ F4 ) ) @ B3 ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F4 ) @ ( complete_Inf_Inf @ ( set @ F4 ) @ ( image2 @ E @ ( set @ F4 ) @ A4 @ C3 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ F4 )
@ ( image2 @ E @ ( set @ F4 )
@ ^ [X3: E] : ( minus_minus @ ( set @ F4 ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_3064_INT__extend__simps_I4_J,axiom,
! [G3: $tType,H8: $tType,C3: set @ H8,A4: set @ G3,B3: H8 > ( set @ G3 )] :
( ( ( C3
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B3 @ C3 ) ) )
= A4 ) )
& ( ( C3
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X3: H8] : ( minus_minus @ ( set @ G3 ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_3065_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A3: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ( ord_less_eq @ A @ A3 @ X2 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X7 )
=> ( ord_less_eq @ A @ Y2 @ X5 ) )
=> ( ord_less_eq @ A @ Y2 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_3066_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,Z2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ).
% cInf_greatest
thf(fact_3067_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A4 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_3068_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Z2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X7 ) @ Z2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ X7 )
& ( ord_less @ A @ X2 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_3069_Inf__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Inf_finite_empty
thf(fact_3070_Sup__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Sup_finite_empty
thf(fact_3071_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,X: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( ( F2 @ I2 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_3072_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A3: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B3 ) @ A3 )
= ( top_top @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( ( sup_sup @ A @ X3 @ A3 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_3073_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B3: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B3 )
=> ( member @ A @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_3074_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B3: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B3 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( inf_inf @ A @ A3 ) @ B3 ) ) ) ) ).
% inf_Sup
thf(fact_3075_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > A,X7: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F2 @ X7 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ X7 )
=> ( ( comp @ A @ C @ B @ G1 @ F2 @ X2 )
= ( comp @ A @ C @ B @ G22 @ F2 @ X2 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_3076_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ ( image2 @ B @ A @ F2 @ B3 ) ) ) ).
% image_Int_subset
thf(fact_3077_Inf__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ).
% Inf_union_distrib
thf(fact_3078_Inter__Un__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A4 @ B3 ) ) ) ).
% Inter_Un_subset
thf(fact_3079_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( image2 @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_3080_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( image2 @ A @ B @ Abs @ A4 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_3081_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A,X: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ X ) )
=> ( ! [Y2: A] :
( ! [I7: B] :
( ( member @ B @ I7 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I7 ) @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_3082_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B3: set @ C,F2: B > A,G2: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B3 ) ) ) ) ) ).
% SUP_mono
thf(fact_3083_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_3084_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G2: B > A,A4: set @ B] :
( ! [X2: B] : ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% SUP_mono'
thf(fact_3085_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% SUP_upper
thf(fact_3086_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_3087_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,U: A,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_3088_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_3089_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F2: B > A,A4: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X3 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_3090_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,C2: B] :
( ( member @ A @ X @ A4 )
=> ( ( image2 @ A @ B
@ ^ [X3: A] : C2
@ A4 )
= ( insert2 @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_3091_image__constant__conv,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X3: B] : C2
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X3: B] : C2
@ A4 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_3092_nat__seg__image__imp__finite,axiom,
! [A: $tType,A4: set @ A,F2: nat > A,N: nat] :
( ( A4
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% nat_seg_image_imp_finite
thf(fact_3093_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
? [N2: nat,F: nat > A] :
( A6
= ( image2 @ nat @ A @ F
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_3094_Inter__insert,axiom,
! [A: $tType,A3: set @ A,B3: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) ) ).
% Inter_insert
thf(fact_3095_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G2: B > A,F2: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X3: B] : ( F2 @ ( G2 @ X3 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_3096_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B3: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ B3 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B4: B] : ( inf_inf @ A @ ( F2 @ B4 ) @ A3 )
@ B3 ) ) ) ) ).
% SUP_inf
thf(fact_3097_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B3 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B4: A] : ( inf_inf @ A @ B4 @ A3 )
@ B3 ) ) ) ) ).
% Sup_inf
thf(fact_3098_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F2: B > A,B3: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ B3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B4: B] : ( inf_inf @ A @ A3 @ ( F2 @ B4 ) )
@ B3 ) ) ) ) ).
% inf_SUP
thf(fact_3099_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A4: set @ B,G2: C > A,B3: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B4: C] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G2 @ B4 ) )
@ B3 ) )
@ A4 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_3100_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,H3: B > A,G2: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H3 @ S )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A @ H3
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G2 @ X3 )
= Y3 ) ) ) )
@ ( image2 @ B @ C @ G2 @ S ) ) ) ) ) ).
% sum.image_gen
thf(fact_3101_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,G2: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_3102_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I4: set @ B,A4: B > A] :
( ( member @ B @ K @ I4 )
=> ( ( sup_sup @ A @ ( A4 @ K ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A4 @ I4 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A4 @ I4 ) ) ) ) ) ).
% SUP_absorb
thf(fact_3103_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,H3: B > A,G2: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H3 @ S )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A @ H3
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G2 @ X3 )
= Y3 ) ) ) )
@ ( image2 @ B @ C @ G2 @ S ) ) ) ) ) ).
% prod.image_gen
thf(fact_3104_UN__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( bot_bot @ ( set @ A ) )
@ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_3105_UN__empty,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_3106_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_3107_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( B3 @ X3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_3108_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G2: C > ( set @ B ),X: C,X7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( G2 @ X ) ) ) @ X7 )
= ( ord_less_eq @ ( set @ B ) @ ( G2 @ X )
@ ( collect @ B
@ ^ [X3: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X3 ) @ X7 ) ) ) ) ).
% UN_image_subset
thf(fact_3109_UN__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ A,F2: A > ( set @ B ),G2: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G2 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_3110_UN__least,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: A > ( set @ B ),C3: set @ B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ X2 ) @ C3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) @ C3 ) ) ).
% UN_least
thf(fact_3111_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_3112_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) @ B3 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( ord_less_eq @ ( set @ A ) @ ( A4 @ X3 ) @ B3 ) ) ) ) ).
% UN_subset_iff
thf(fact_3113_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A4: set @ A,A3: B,B3: A > ( set @ B )] :
( ( member @ A @ U @ A4 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( insert2 @ B @ A3 @ ( B3 @ X3 ) )
@ A4 ) )
= ( insert2 @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B3 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_3114_UN__extend__simps_I5_J,axiom,
! [I6: $tType,J4: $tType,A4: set @ I6,B3: J4 > ( set @ I6 ),C3: set @ J4] :
( ( inf_inf @ ( set @ I6 ) @ A4 @ ( complete_Sup_Sup @ ( set @ I6 ) @ ( image2 @ J4 @ ( set @ I6 ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ I6 )
@ ( image2 @ J4 @ ( set @ I6 )
@ ^ [X3: J4] : ( inf_inf @ ( set @ I6 ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) ) ) ).
% UN_extend_simps(5)
thf(fact_3115_UN__extend__simps_I4_J,axiom,
! [H8: $tType,G3: $tType,A4: G3 > ( set @ H8 ),C3: set @ G3,B3: set @ H8] :
( ( inf_inf @ ( set @ H8 ) @ ( complete_Sup_Sup @ ( set @ H8 ) @ ( image2 @ G3 @ ( set @ H8 ) @ A4 @ C3 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ H8 )
@ ( image2 @ G3 @ ( set @ H8 )
@ ^ [X3: G3] : ( inf_inf @ ( set @ H8 ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) ) ) ).
% UN_extend_simps(4)
thf(fact_3116_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B3: set @ A,A4: B > ( set @ A ),I4: set @ B] :
( ( inf_inf @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( inf_inf @ ( set @ A ) @ B3 @ ( A4 @ I3 ) )
@ I4 ) ) ) ).
% Int_UN_distrib
thf(fact_3117_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B,B3: C > ( set @ A ),J5: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B3 @ J5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( inf_inf @ ( set @ A ) @ ( A4 @ I3 ) @ ( B3 @ J3 ) )
@ J5 ) )
@ I4 ) ) ) ).
% Int_UN_distrib2
thf(fact_3118_UN__absorb,axiom,
! [B: $tType,A: $tType,K: A,I4: set @ A,A4: A > ( set @ B )] :
( ( member @ A @ K @ I4 )
=> ( ( sup_sup @ ( set @ B ) @ ( A4 @ K ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) ) ) ).
% UN_absorb
thf(fact_3119_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),B3: B > ( set @ A ),I4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( sup_sup @ ( set @ A ) @ ( A4 @ I3 ) @ ( B3 @ I3 ) )
@ I4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ I4 ) ) ) ) ).
% UN_Un_distrib
thf(fact_3120_Un__Union__image,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),B3: B > ( set @ A ),C3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( sup_sup @ ( set @ A ) @ ( A4 @ X3 ) @ ( B3 @ X3 ) )
@ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ C3 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Un_Union_image
thf(fact_3121_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K6: $tType,A4: K6 > ( set @ L5 ),C3: set @ K6,B3: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image2 @ K6 @ ( set @ L5 ) @ A4 @ C3 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image2 @ K6 @ ( set @ L5 )
@ ^ [X3: K6] : ( minus_minus @ ( set @ L5 ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) ) ) ).
% UN_extend_simps(6)
thf(fact_3122_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_3123_vimage__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,B3: C > ( set @ B ),A4: set @ C] :
( ( vimage @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( vimage @ A @ B @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% vimage_UN
thf(fact_3124_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B,X: A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ( F2 @ Y2 )
= ( F2 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_3125_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_3126_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_3127_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,A3: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X7 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ( ord_less @ A @ A3 @ X3 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_3128_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Inf_le_Sup
thf(fact_3129_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,M4: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M4 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ M4 ) ) ) ) ).
% cSUP_least
thf(fact_3130_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,C2: A,F2: B > A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I2 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= C2 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( ( F2 @ X3 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3131_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( member @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ A4 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_3132_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_3133_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_3134_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: A,X: B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F2 @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_3135_Union__image__empty,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3136_Union__image__insert,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A3: B,B3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( insert2 @ B @ A3 @ B3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ B3 ) ) ) ) ).
% Union_image_insert
thf(fact_3137_finite__vimageD,axiom,
! [A: $tType,B: $tType,H3: A > B,F5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H3 @ F5 ) )
=> ( ( ( image2 @ A @ B @ H3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimageD
thf(fact_3138_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B3: set @ B,F2: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B3 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_3139_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_3140_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A4 ) )
= ( bot_bot @ A ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_3141_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T2: set @ C,G2: B > C,H3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G2 @ S ) @ T2 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y3: C] :
( groups7311177749621191930dd_sum @ B @ A @ H3
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G2 @ X3 )
= Y3 ) ) ) )
@ T2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H3 @ S ) ) ) ) ) ) ).
% sum.group
thf(fact_3142_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: B,A4: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ A @ ( F2 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% SUP_insert
thf(fact_3143_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ C,G2: B > C,H3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G2 @ S ) @ T2 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y3: C] :
( groups7121269368397514597t_prod @ B @ A @ H3
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G2 @ X3 )
= Y3 ) ) ) )
@ T2 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ S ) ) ) ) ) ) ).
% prod.group
thf(fact_3144_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M4: B > A,A4: set @ B,B3: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M4 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M4 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M4 @ B3 ) ) ) ) ) ).
% SUP_union
thf(fact_3145_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ B,A3: A,B3: B > ( set @ A )] :
( ( ( C3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ C3 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ A3 @ ( B3 @ X3 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_3146_UN__extend__simps_I3_J,axiom,
! [E: $tType,F4: $tType,C3: set @ F4,A4: set @ E,B3: F4 > ( set @ E )] :
( ( ( C3
= ( bot_bot @ ( set @ F4 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A4 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F4 @ ( set @ E ) @ B3 @ C3 ) ) )
= A4 ) )
& ( ( C3
!= ( bot_bot @ ( set @ F4 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A4 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F4 @ ( set @ E ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F4 @ ( set @ E )
@ ^ [X3: F4] : ( sup_sup @ ( set @ E ) @ A4 @ ( B3 @ X3 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3147_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C3: set @ C,A4: C > ( set @ D ),B3: set @ D] :
( ( ( C3
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C3 ) ) @ B3 )
= B3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C3 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X3: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X3 ) @ B3 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3148_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: A > C,G2: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F2 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G2 )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_3149_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ L ) ) @ E4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E4 ) ) ) ) ).
% cInf_asclose
thf(fact_3150_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,H3: B > C,G2: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B,Y2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( member @ B @ Y2 @ A4 )
=> ( ( X2 != Y2 )
=> ( ( ( H3 @ X2 )
= ( H3 @ Y2 ) )
=> ( ( G2 @ ( H3 @ X2 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G2 @ H3 ) @ A4 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_3151_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_3152_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_3153_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F2 @ A4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A4 ) ) ) ).
% finite_vimageD'
thf(fact_3154_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F: A > B,B5: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( vimage @ A @ B @ F @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B5 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3155_UN__UN__finite__eq,axiom,
! [A: $tType,A4: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N2: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_3156_UN__le__add__shift,axiom,
! [A: $tType,M4: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M4 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M4 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_3157_UN__le__add__shift__strict,axiom,
! [A: $tType,M4: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M4 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M4 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_3158_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_3159_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3160_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3161_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,A4: B > ( set @ C ),G2: C > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( finite_finite2 @ C @ ( A4 @ X2 ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A4 @ X2 ) @ ( A4 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( A4 @ X3 ) )
@ I4 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_3162_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,A4: B > ( set @ C ),G2: C > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( finite_finite2 @ C @ ( A4 @ X2 ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A4 @ X2 ) @ ( A4 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( A4 @ X3 ) )
@ I4 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_3163_card__UN__le,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B )] :
( ( finite_finite2 @ A @ I4 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A4 @ I3 ) )
@ I4 ) ) ) ).
% card_UN_le
thf(fact_3164_UN__finite__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),C3: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C3 ) ) ).
% UN_finite_subset
thf(fact_3165_UN__finite2__eq,axiom,
! [A: $tType,A4: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_3166_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F4: $tType,Rep: A > B,Abs: B > A,S: B > ( set @ D ),M4: C > B,F2: E > D,S4: C > ( set @ E ),G2: F4 > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ( comp @ B @ ( set @ D ) @ C @ S @ M4 )
= ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F2 ) @ S4 ) )
=> ( ( comp @ A @ ( set @ D ) @ F4 @ ( comp @ B @ ( set @ D ) @ A @ S @ Rep ) @ ( comp @ C @ A @ F4 @ ( comp @ B @ A @ C @ Abs @ M4 ) @ G2 ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F4 @ ( image2 @ E @ D @ F2 ) @ ( comp @ C @ ( set @ E ) @ F4 @ S4 @ G2 ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_3167_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_3168_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_3169_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_3170_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_3171_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_3172_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_3173_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R: set @ B,G2: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G2 @ S ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X3: A] : ( F2 @ ( G2 @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y3: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ S )
& ( ( G2 @ X3 )
= Y3 ) ) ) ) )
@ ( F2 @ Y3 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3174_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B )] :
( ( finite_finite2 @ A @ I4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( finite_finite2 @ B @ ( A4 @ X2 ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A4 @ X2 ) @ ( A4 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A4 @ I3 ) )
@ I4 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_3175_UN__finite2__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_3176_UN__le__eq__Un0,axiom,
! [A: $tType,M4: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M4 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M4 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_3177_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_3178_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_3179_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( image2 @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_3180_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F2: B > A,B3: set @ A,A4: set @ B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ B3 ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3181_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F2 @ A4 )
= ( bot_bot @ ( set @ B ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_3182_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: multiset @ A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X3: B] : C2
@ A4 ) )
= C2 ) ) ).
% subset_mset.cINF_const
thf(fact_3183_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: multiset @ A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X3: B] : C2
@ A4 ) )
= C2 ) ) ).
% subset_mset.cSUP_const
thf(fact_3184_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A4 )
=> ( member @ C @ ( F2 @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_3185_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X3: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F: A > B] : ( F @ X3 )
@ A6 ) ) ) ) ) ).
% Sup_apply
thf(fact_3186_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X3: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F: A > B] : ( F @ X3 )
@ A6 ) ) ) ) ) ).
% Inf_apply
thf(fact_3187_Inf__int__def,axiom,
( ( complete_Inf_Inf @ int )
= ( ^ [X4: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image2 @ int @ int @ ( uminus_uminus @ int ) @ X4 ) ) ) ) ) ).
% Inf_int_def
thf(fact_3188_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R3 @ I3 ) )
@ S ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_3189_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S8 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_3190_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ A
@ ^ [X3: A] : ( complete_Inf_Inf @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X3 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_3191_INF__Int__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I3: set @ A,X3: A] : ( member @ A @ X3 @ I3 )
@ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_3192_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R3 @ I3 ) )
@ S ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_3193_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
@ S ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% INF_Int_eq2
thf(fact_3194_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S8 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_3195_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B3: set @ I6,F5: I6 > ( filter @ A )] :
( ! [X8: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X8 @ B3 )
=> ( ( finite_finite2 @ I6 @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F5 @ X8 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F5 @ B3 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_3196_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R3: B > ( set @ A ),S: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I3: B,X3: A] : ( member @ A @ X3 @ ( R3 @ I3 ) )
@ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R3 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_3197_Inf__nat__def1,axiom,
! [K5: set @ nat] :
( ( K5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_3198_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X3: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F: A > B] : ( F @ X3 )
@ A6 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_3199_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A6: set @ ( A > B ),X3: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F: A > B] : ( F @ X3 )
@ A6 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_3200_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
@ S ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_3201_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ A
@ ^ [X3: A] : ( complete_Sup_Sup @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_3202_SUP__Sup__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I3: set @ A,X3: A] : ( member @ A @ X3 @ I3 )
@ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_3203_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R3: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I3: B,X3: A] : ( member @ A @ X3 @ ( R3 @ I3 ) )
@ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R3 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_3204_Int__Union2,axiom,
! [A: $tType,B3: set @ ( set @ A ),A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) @ A4 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [C7: set @ A] : ( inf_inf @ ( set @ A ) @ C7 @ A4 )
@ B3 ) ) ) ).
% Int_Union2
thf(fact_3205_Int__Union,axiom,
! [A: $tType,A4: set @ A,B3: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 ) @ B3 ) ) ) ).
% Int_Union
thf(fact_3206_Un__Inter,axiom,
! [A: $tType,A4: set @ A,B3: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 ) @ B3 ) ) ) ).
% Un_Inter
thf(fact_3207_vimage__Union,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ ( set @ B )] :
( ( vimage @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ A4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F2 ) @ A4 ) ) ) ).
% vimage_Union
thf(fact_3208_in__image__insert__iff,axiom,
! [A: $tType,B3: set @ ( set @ A ),X: A,A4: set @ A] :
( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ B3 )
=> ~ ( member @ A @ X @ C8 ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X ) @ B3 ) )
= ( ( member @ A @ X @ A4 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_3209_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_3210_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A4: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= A4 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 ) @ B11 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_3211_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A4: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A4 )
= A4 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A4 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A4 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_3212_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_3213_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: $o > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ $o @ A @ A4 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_3214_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: $o > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ $o @ A @ A4 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ A @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_3215_UN__bool__eq,axiom,
! [A: $tType,A4: $o > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A4 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ ( set @ A ) @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_3216_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ $o @ ( set @ A )
@ ^ [B4: $o] : ( if @ ( set @ A ) @ B4 @ A6 @ B5 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_3217_INT__bool__eq,axiom,
! [A: $tType,A4: $o > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A4 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ ( set @ A ) @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_3218_image__Suc__atMost,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_3219_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_3220_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X3: int] : ( plus_plus @ int @ X3 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_3221_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image2 @ code_integer @ code_integer
@ ^ [X3: code_integer] : ( plus_plus @ code_integer @ X3 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_3222_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_3223_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_3224_surjD,axiom,
! [A: $tType,B: $tType,F2: B > A,Y: A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_3225_surjE,axiom,
! [A: $tType,B: $tType,F2: B > A,Y: A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_3226_surjI,axiom,
! [B: $tType,A: $tType,G2: B > A,F2: A > B] :
( ! [X2: A] :
( ( G2 @ ( F2 @ X2 ) )
= X2 )
=> ( ( image2 @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_3227_surj__def,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y3: A] :
? [X3: B] :
( Y3
= ( F2 @ X3 ) ) ) ) ).
% surj_def
thf(fact_3228_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S3: set @ A,T4: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S3 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T4 ) ) ) ) ).
% translation_Int
thf(fact_3229_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: A > C] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ C @ G2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image2 @ B @ C @ ( comp @ A @ C @ B @ G2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_3230_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T4: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T4 ) ) ) ) ).
% translation_Compl
thf(fact_3231_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3232_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S3: set @ A,T4: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ S3 )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T4 ) ) ) ) ).
% translation_subtract_Int
thf(fact_3233_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S3: set @ A,T4: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ S3 )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T4 ) ) ) ) ).
% translation_subtract_diff
thf(fact_3234_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T4: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T4 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T4 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_3235_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F2: A > B,V: D > A] :
( ( image2 @ D @ B @ ( comp @ A @ B @ D @ F2 @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image2 @ A @ B @ F2 @ ( image2 @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_3236_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F2: A > B,G2: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F2 @ X )
= ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_3237_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,F2: A > B,G2: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F2 @ X )
= ( comp @ A @ B @ D @ G2 @ X ) ) ) ).
% fun.map_cong0
thf(fact_3238_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F2: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image2 @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F2 @ X )
= ( comp @ A @ B @ D @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_3239_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image2 @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_3240_subset__mset_OcInf__singleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.cInf_singleton
thf(fact_3241_subset__mset_OcSup__singleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.cSup_singleton
thf(fact_3242_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: A,A4: set @ A,B3: A > B > C > $o,B2: B,C2: C] :
( ( member @ A @ A3 @ A4 )
=> ( ( B3 @ A3 @ B2 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B3 @ A4 ) @ B2 @ C2 ) ) ) ).
% SUP2_I
thf(fact_3243_INF1__I,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: A > B > $o,B2: B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( B3 @ X2 @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B3 @ A4 ) @ B2 ) ) ).
% INF1_I
thf(fact_3244_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B3: A > B > C > $o,B2: B,C2: C] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( B3 @ X2 @ B2 @ C2 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B3 @ A4 ) @ B2 @ C2 ) ) ).
% INF2_I
thf(fact_3245_SUP1__I,axiom,
! [A: $tType,B: $tType,A3: A,A4: set @ A,B3: A > B > $o,B2: B] :
( ( member @ A @ A3 @ A4 )
=> ( ( B3 @ A3 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B3 @ A4 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_3246_Sup__multiset__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Sup_multiset_empty
thf(fact_3247_SUP1__E,axiom,
! [B: $tType,A: $tType,B3: B > A > $o,A4: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B3 @ A4 ) @ B2 )
=> ~ ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ~ ( B3 @ X2 @ B2 ) ) ) ).
% SUP1_E
thf(fact_3248_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B3: C > A > B > $o,A4: set @ C,B2: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B3 @ A4 ) @ B2 @ C2 )
=> ~ ! [X2: C] :
( ( member @ C @ X2 @ A4 )
=> ~ ( B3 @ X2 @ B2 @ C2 ) ) ) ).
% SUP2_E
thf(fact_3249_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B3: C > A > B > $o,A4: set @ C,B2: A,C2: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B3 @ A4 ) @ B2 @ C2 )
=> ( ~ ( B3 @ A3 @ B2 @ C2 )
=> ~ ( member @ C @ A3 @ A4 ) ) ) ).
% INF2_E
thf(fact_3250_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B3: C > A > B > $o,A4: set @ C,B2: A,C2: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B3 @ A4 ) @ B2 @ C2 )
=> ( ( member @ C @ A3 @ A4 )
=> ( B3 @ A3 @ B2 @ C2 ) ) ) ).
% INF2_D
thf(fact_3251_INF1__E,axiom,
! [A: $tType,B: $tType,B3: B > A > $o,A4: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B3 @ A4 ) @ B2 )
=> ( ~ ( B3 @ A3 @ B2 )
=> ~ ( member @ B @ A3 @ A4 ) ) ) ).
% INF1_E
thf(fact_3252_INF1__D,axiom,
! [B: $tType,A: $tType,B3: B > A > $o,A4: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B3 @ A4 ) @ B2 )
=> ( ( member @ B @ A3 @ A4 )
=> ( B3 @ A3 @ B2 ) ) ) ).
% INF1_D
thf(fact_3253_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I4: set @ A,F5: A > ( filter @ B )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ I4 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I2 ) @ ( F5 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I4 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ I4 )
& ( ( F5 @ X3 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_3254_Inf__multiset__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( multiset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Inf_multiset_empty
thf(fact_3255_Inf__filter__not__bot,axiom,
! [A: $tType,B3: set @ ( filter @ A )] :
( ! [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B3 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X8 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B3 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_3256_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T4: D > A] :
( ( comp @ A @ A @ D
@ ^ [X3: A] : X3
@ T4 )
= T4 ) ).
% fun.map_ident
thf(fact_3257_UN__UN__split__split__eq,axiom,
! [A: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A4: B > C > D > E > ( set @ A ),Y4: set @ ( product_prod @ D @ E ),X7: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X23: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A ) @ ( product_case_prod @ D @ E @ ( set @ A ) @ ( A4 @ X12 @ X23 ) ) @ Y4 ) ) )
@ X7 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X3: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A )
@ ^ [Y3: product_prod @ D @ E] :
( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X23: C] : ( product_case_prod @ D @ E @ ( set @ A ) @ ( A4 @ X12 @ X23 ) @ Y3 )
@ X3 )
@ Y4 ) )
@ X7 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_3258_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F: A > nat,R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( ord_less @ nat @ ( F @ X3 ) @ ( F @ Y3 ) )
| ( ( ord_less_eq @ nat @ ( F @ X3 ) @ ( F @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_3259_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A3: A,A4: set @ A,F2: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A3 @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( F2 @ X2 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ A4 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_3260_fold__union__pair,axiom,
! [B: $tType,A: $tType,B3: set @ A,X: B,A4: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B3 ) )
@ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) )
@ A4
@ B3 ) ) ) ).
% fold_union_pair
thf(fact_3261_range__nat__of__char,axiom,
( ( image2 @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_3262_fold__empty,axiom,
! [B: $tType,A: $tType,F2: B > A > A,Z2: A] :
( ( finite_fold @ B @ A @ F2 @ Z2 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ).
% fold_empty
thf(fact_3263_union__fold__insert,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ B3 @ A4 ) ) ) ).
% union_fold_insert
thf(fact_3264_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B3 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B3 @ A4 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_3265_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ B3 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B3 @ A4 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_3266_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( complete_Sup_Sup @ A @ A4 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A4 ) ) ) ) ).
% Sup_fold_sup
thf(fact_3267_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( complete_Inf_Inf @ A @ A4 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A4 ) ) ) ) ).
% Inf_fold_inf
thf(fact_3268_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_3269_image__fold__insert,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( image2 @ A @ B @ F2 @ A4 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K4: A] : ( insert2 @ B @ ( F2 @ K4 ) )
@ ( bot_bot @ ( set @ B ) )
@ A4 ) ) ) ).
% image_fold_insert
thf(fact_3270_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B3: A,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( sup_sup @ A @ B3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ B3 @ A4 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_3271_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B3: A,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( inf_inf @ A @ B3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ B3 @ A4 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_3272_Id__on__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( id_on @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A4 ) ) ) ).
% Id_on_fold
thf(fact_3273_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ ( bot_bot @ A ) @ A4 ) ) ) ) ).
% SUP_fold_sup
thf(fact_3274_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ ( top_top @ A ) @ A4 ) ) ) ) ).
% INF_fold_inf
thf(fact_3275_mlex__leq,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ) ).
% mlex_leq
thf(fact_3276_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_3277_mlex__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ).
% mlex_less
thf(fact_3278_Set__filter__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( filter3 @ A @ P @ A4 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X3: A,A11: set @ A] : ( if @ ( set @ A ) @ ( P @ X3 ) @ ( insert2 @ A @ X3 @ A11 ) @ A11 )
@ ( bot_bot @ ( set @ A ) )
@ A4 ) ) ) ).
% Set_filter_fold
thf(fact_3279_pred__def,axiom,
( pred2
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_3280_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P2: A > $o,A6: set @ A] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A6 )
& ( P2 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_3281_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_3282_inter__Set__filter,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( filter3 @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 )
@ B3 ) ) ) ).
% inter_Set_filter
thf(fact_3283_in__measure,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_3284_in__finite__psubset,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B3 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
& ( finite_finite2 @ A @ B3 ) ) ) ).
% in_finite_psubset
thf(fact_3285_Pow__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pow2 @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X3 ) @ A6 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A4 ) ) ) ).
% Pow_fold
thf(fact_3286_relpow__finite__bounded1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_3287_range__abs__Nats,axiom,
( ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_3288_relpow__1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R )
= R ) ).
% relpow_1
thf(fact_3289_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_3290_Pow__Int__eq,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B3 ) ) ) ).
% Pow_Int_eq
thf(fact_3291_Pow__empty,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_3292_Pow__singleton__iff,axiom,
! [A: $tType,X7: set @ A,Y4: set @ A] :
( ( ( pow2 @ A @ X7 )
= ( insert2 @ ( set @ A ) @ Y4 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
& ( Y4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_3293_Pow__bottom,axiom,
! [A: $tType,B3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B3 ) ) ).
% Pow_bottom
thf(fact_3294_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A ),X5: A,Y6: A,Z8: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z8 ) @ R ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ W ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z8 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_3295_Pow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_3296_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% Pow_def
thf(fact_3297_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_3298_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_3299_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( 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
thf(fact_3300_relpow__0__I,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ).
% relpow_0_I
thf(fact_3301_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 ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I2
thf(fact_3302_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_3303_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ? [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_3304_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 ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I
thf(fact_3305_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_3306_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( pow2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X3: B] : ( pow2 @ A @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% Pow_INT_eq
thf(fact_3307_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( 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
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_3308_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( 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
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_3309_relpow__empty,axiom,
! [A: $tType,N: nat] :
( ( 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
thf(fact_3310_Un__Pow__subset,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B3 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% Un_Pow_subset
thf(fact_3311_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X3: B] : ( pow2 @ A @ ( B3 @ X3 ) )
@ A4 ) )
@ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) ) ) ).
% UN_Pow_subset
thf(fact_3312_Pow__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( pow2 @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ A3 ) @ ( pow2 @ A @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_3313_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
= ( ? [F: nat > A] :
( ( ( F @ ( zero_zero @ nat ) )
= A3 )
& ( ( F @ N )
= B2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_3314_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_3315_binomial__def,axiom,
( binomial
= ( ^ [N2: nat,K4: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K7: set @ nat] :
( ( member @ ( set @ nat ) @ K7 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
& ( ( finite_card @ nat @ K7 )
= K4 ) ) ) ) ) ) ).
% binomial_def
thf(fact_3316_relpow__finite__bounded,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_3317_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A6: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B5 )
& ( finite_finite2 @ A @ B5 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_3318_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N2: nat,R2: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I3 @ R2 )
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I3 )
& ( ord_less_eq @ nat @ I3 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_3319_trancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_3320_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_3321_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A4: set @ A,B3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( finite_fold @ A @ B @ F2 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A4 ) @ B3 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_3322_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_rtrancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_3323_trancl__empty,axiom,
! [A: $tType] :
( ( transitive_trancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trancl_empty
thf(fact_3324_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_3325_tranclD,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% tranclD
thf(fact_3326_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtranclD
thf(fact_3327_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R ) ) ) ).
% tranclD2
thf(fact_3328_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_3329_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_3330_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_3331_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_3332_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_3333_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_3334_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ! [A8: A,B7: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A8 @ B7 ) ) @ R3 )
=> ( P @ A8 @ B7 ) )
=> ( ! [A8: A,B7: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A8 @ B7 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R3 )
=> ( ( P @ A8 @ B7 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_3335_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R3 )
=> ~ ! [B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B7 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_3336_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ? [A5: A,B4: A] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R3 ) )
| ? [A5: A,B4: A,C5: A] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ ( transitive_trancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_3337_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_3338_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ~ ! [C4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C4 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C4 @ B2 ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_3339_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_3340_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y2 ) @ R3 )
=> ( P @ Y2 ) )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_3341_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_3342_converse__tranclE,axiom,
! [A: $tType,X: A,Z2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_3343_irrefl__trancl__rD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_3344_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_3345_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_3346_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ( P @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ X2 @ Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ Y2 @ Z3 )
=> ( P @ X2 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_3347_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B2 ) @ R3 )
=> ( P @ Y2 ) )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_3348_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A8: A,B7: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A8 @ B7 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R3 )
=> ( ( P @ A8 @ B7 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_3349_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A8: A,B7: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A8 @ B7 ) ) @ R3 )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_3350_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A8: A,B7: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R3 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A8 @ B7 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_3351_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( 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 ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_3352_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A22 != A1 )
=> ~ ! [B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B7 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_3353_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( ? [A5: A] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: A,B4: A,C5: A] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_3354_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_3355_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_3356_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A3 != B2 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B2 ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_3357_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_3358_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ A3 )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_3359_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( X != Z2 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_3360_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ B2 )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_3361_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_3362_trancl__union__outside,axiom,
! [A: $tType,V: A,W2: A,E3: set @ ( product_prod @ A @ A ),U4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E3 @ U4 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E3 ) )
=> ? [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E3 @ U4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ U4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E3 @ U4 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_3363_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W2: A,V1: A,V22: A,E3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E3 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ E3 ) )
=> ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W2 ) @ ( transitive_rtrancl @ A @ E3 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_3364_in__rtrancl__UnI,axiom,
! [A: $tType,X: product_prod @ A @ A,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R ) )
| ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ S ) ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_3365_rtrancl__Un__rtrancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) )
= ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_3366_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q2 ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ Q2 )
=> ( Y2 = X2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_3367_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q2 ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ Q2 )
=> ( X2 = Y2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_3368_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,A4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_3369_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W2: A,E3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E3 @ R ) ) )
=> ~ ( member @ A @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_3370_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W2: A,E3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E3 @ R ) ) )
=> ~ ( member @ A @ W2 @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_3371_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B4 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_3372_empty__in__Fpow,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) ) ).
% empty_in_Fpow
thf(fact_3373_rtrancl__Un__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ).
% rtrancl_Un_subset
thf(fact_3374_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E3: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A3 ) @ ( F2 @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F2 ) @ E3 ) ) ) ) ).
% rtrancl_mapI
thf(fact_3375_Fpow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_Fpow @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_3376_finite__trancl__ntranl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R ) @ ( one_one @ nat ) ) @ R ) ) ) ).
% finite_trancl_ntranl
thf(fact_3377_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_trancl @ A @ R3 ) )
| ( X3 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
| ( Y3 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_3378_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [X4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ A6 )
& ( finite_finite2 @ A @ X4 ) ) ) ) ) ).
% Fpow_def
thf(fact_3379_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_3380_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ A,F2: A > B > B,G2: C > A,R: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) ) @ S )
=> ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G2 ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_3381_rtrancl__is__UN__relpow,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_is_UN_relpow
thf(fact_3382_rtrancl__imp__UN__relpow,axiom,
! [A: $tType,P4: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ P4
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_imp_UN_relpow
thf(fact_3383_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A4: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_3384_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A4: set @ A,X: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ A4 )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_3385_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_3386_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A4: B > ( set @ A ),I: B,B3: set @ A,J5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A4 @ I @ B3 ) @ J5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ B ) @ J5 @ ( insert2 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J5 ) @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_3387_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3388_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_3389_rat__minus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B4 @ C5 ) ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_3390_bit__0__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) )
= ( bot_bot @ ( nat > $o ) ) ) ) ).
% bit_0_eq
thf(fact_3391_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_3392_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_3393_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_3394_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_3395_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_3396_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(4)
thf(fact_3397_bit__minus__numeral__int_I1_J,axiom,
! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_3398_bit__minus__numeral__int_I2_J,axiom,
! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_3399_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_3400_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q5: rat,R4: rat] : ( plus_plus @ rat @ Q5 @ ( uminus_uminus @ rat @ R4 ) ) ) ) ).
% diff_rat_def
thf(fact_3401_quotient__of__div,axiom,
! [R3: rat,N: int,D3: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ N @ D3 ) )
=> ( R3
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% quotient_of_div
thf(fact_3402_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X2: nat,Y2: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X2 @ Y2 ) ) ) ).
% eq_Abs_Integ
thf(fact_3403_finite__update__induct,axiom,
! [B: $tType,A: $tType,F2: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( F2 @ A5 )
!= C2 ) ) )
=> ( ( P
@ ^ [A5: A] : C2 )
=> ( ! [A8: A,B7: B,F3: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C5: A] :
( ( F3 @ C5 )
!= C2 ) ) )
=> ( ( ( F3 @ A8 )
= C2 )
=> ( ( B7 != C2 )
=> ( ( P @ F3 )
=> ( P @ ( fun_upd @ A @ B @ F3 @ A8 @ B7 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_update_induct
thf(fact_3404_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_3405_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% not_bit_1_Suc
thf(fact_3406_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_3407_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A5 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_3408_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_3409_quotient__of__denom__pos,axiom,
! [R3: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_3410_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W2 ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_3411_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W2 ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_3412_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_3413_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_3414_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_3415_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_3416_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P6: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P6 ) ) ) ) ).
% rat_floor_code
thf(fact_3417_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_3418_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_3419_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P6: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C5 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_3420_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C5 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_3421_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_3422_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M7: nat] :
( ( ord_less_eq @ nat @ N3 @ M7 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M7 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3423_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A4: set @ B,F2: B > A,Y: A] :
( ( ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y ) @ A4 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y ) @ A4 )
= ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3424_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_3425_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_3426_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_3427_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_3428_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_3429_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_3430_rat__divide__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C5 @ B4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_3431_rat__times__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B4 ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_3432_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( bit_se4730199178511100633sh_bit @ A @ K4 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% take_bit_sum
thf(fact_3433_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_3434_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_3435_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_3436_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A5 ) @ N2 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3437_rat__plus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B4 @ C5 ) ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_3438_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_3439_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: A > $o,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R ) )
= ( same_fst @ A @ B @ P
@ ^ [X3: A] : ( transitive_trancl @ B @ ( R @ X3 ) ) ) ) ).
% same_fst_trancl
thf(fact_3440_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N2: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_3441_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B4 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_3442_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A7 @ B6 ) ) @ ( lex_prod @ A @ B @ R3 @ S3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A7 ) @ R3 )
| ( ( A3 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B6 ) @ S3 ) ) ) ) ).
% in_lex_prod
thf(fact_3443_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_3444_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( ( inverse_inverse @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_3445_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3446_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% inverse_minus_eq
thf(fact_3447_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_3448_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_3449_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_3450_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_3451_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_3452_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A5: rat] :
( if @ rat
@ ( A5
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_3453_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X: A] :
( ( ( times_times @ A @ Y @ X )
= ( times_times @ A @ X @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3454_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_3455_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_3456_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_3457_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3458_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% divide_inverse
thf(fact_3459_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A5 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3460_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3461_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3462_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3463_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3464_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3465_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3466_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_3467_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3468_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_3469_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_3470_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_3471_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_3472_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3473_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_3474_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_3475_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3476_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3477_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_3478_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3479_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra2: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A5: A,B4: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A14: A,B12: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A14 ) @ Ra2 )
| ( ( A5 = A14 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_3480_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y7: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_3481_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P2: A > $o,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X9: A,Y8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( X9 = X3 )
& ( P2 @ X3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y3 ) @ ( R2 @ X3 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_3482_char__of__integer__code,axiom,
( char_of_integer
= ( ^ [K4: code_integer] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q0: code_integer,B0: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q1: code_integer,B1: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q22: code_integer,B22: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q32: code_integer,B32: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q42: code_integer,B42: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q52: code_integer,B52: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q62: code_integer,B62: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Uu: code_integer] : ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 )
@ ( code_bit_cut_integer @ Q62 ) )
@ ( code_bit_cut_integer @ Q52 ) )
@ ( code_bit_cut_integer @ Q42 ) )
@ ( code_bit_cut_integer @ Q32 ) )
@ ( code_bit_cut_integer @ Q22 ) )
@ ( code_bit_cut_integer @ Q1 ) )
@ ( code_bit_cut_integer @ Q0 ) )
@ ( code_bit_cut_integer @ K4 ) ) ) ) ).
% char_of_integer_code
thf(fact_3483_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_3484_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X3: rat] :
( the @ int
@ ^ [Z5: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z5 ) @ X3 )
& ( ord_less @ rat @ X3 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3485_Frct__code__post_I9_J,axiom,
! [Q4: product_prod @ int @ int] :
( ( uminus_uminus @ rat @ ( uminus_uminus @ rat @ ( frct @ Q4 ) ) )
= ( frct @ Q4 ) ) ).
% Frct_code_post(9)
thf(fact_3486_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X4: set @ A] :
( the @ A
@ ^ [X3: A] :
( X4
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3487_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_3488_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_3489_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
= ( one_one @ rat ) ) ).
% Frct_code_post(3)
thf(fact_3490_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_3491_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_3492_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_3493_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_3494_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% old.rec_prod_def
thf(fact_3495_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X3: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% old.rec_nat_def
thf(fact_3496_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_3497_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X3: A] : X3 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_3498_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_3499_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_3500_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_3501_the1I2,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( Q2 @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_3502_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P2: $o,X3: A,Y3: A] :
( the @ A
@ ^ [Z5: A] :
( ( P2
=> ( Z5 = X3 ) )
& ( ~ P2
=> ( Z5 = Y3 ) ) ) ) ) ) ).
% If_def
thf(fact_3503_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q2: A > $o] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( Q2 @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_3504_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_3505_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_3506_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X3: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_3507_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X3: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_3508_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X5: A] :
( ( member @ A @ X5 @ S )
& ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_3509_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > A,G2: C > B,A4: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A4 )
@ ^ [X3: A] : ( image2 @ C @ B @ G2 @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3510_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_3511_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B3: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ B @ B2 @ ( B3 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_3512_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
= ( ( member @ A @ A3 @ A4 )
& ( member @ B @ B2 @ ( B3 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_3513_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q2: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] :
( ( P @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_3514_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_3515_Times__empty,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B3
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_3516_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_3517_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A4: set @ A,X7: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : X7 )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_3518_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A4 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_3519_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A4 )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_3520_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( finite_finite2 @ B @ ( B3 @ A8 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) ) ) ) ).
% finite_SigmaI
thf(fact_3521_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_3522_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B,B3: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F2 )
@ ( product_Sigma @ B @ B @ A4
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F2 @ A4 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F2 @ B3 ) ) ) ).
% pairself_image_cart
thf(fact_3523_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B3: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
@ ^ [Uu: A] : ( insert2 @ B @ B2 @ B3 ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( insert2 @ B @ B2 @ B3 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
@ ^ [Uu: A] : B3 ) ) ) ) ).
% insert_Times_insert
thf(fact_3524_card__SigmaI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( finite_finite2 @ B @ ( B3 @ X2 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A5: A] : ( finite_card @ B @ ( B3 @ A5 ) )
@ A4 ) ) ) ) ).
% card_SigmaI
thf(fact_3525_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C3: set @ A,A4: set @ B,B3: set @ B] :
( ( member @ A @ X @ C3 )
=> ( ( ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : C3 )
= ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : C3 ) )
= ( A4 = B3 ) ) ) ).
% Times_eq_cancel2
thf(fact_3526_Sigma__cong,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ A,C3: A > ( set @ B ),D4: A > ( set @ B )] :
( ( A4 = B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D4 @ X2 ) ) )
=> ( ( product_Sigma @ A @ B @ A4 @ C3 )
= ( product_Sigma @ A @ B @ B3 @ D4 ) ) ) ) ).
% Sigma_cong
thf(fact_3527_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ! [Y2: B] :
( ( member @ B @ Y2 @ ( B3 @ X2 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_3528_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
=> ( member @ A @ A3 @ A4 ) ) ).
% SigmaD1
thf(fact_3529_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
=> ( member @ B @ B2 @ ( B3 @ A3 ) ) ) ).
% SigmaD2
thf(fact_3530_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B3: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
=> ~ ( ( member @ A @ A3 @ A4 )
=> ~ ( member @ B @ B2 @ ( B3 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_3531_times__eq__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B,C3: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
= ( product_Sigma @ A @ B @ C3
@ ^ [Uu: A] : D4 ) )
= ( ( ( A4 = C3 )
& ( B3 = D4 ) )
| ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B3
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C3
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_3532_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ A,C3: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A4 @ B3 )
@ ^ [Uu: A] : C3 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C3 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : C3 ) ) ) ).
% Times_Diff_distrib1
thf(fact_3533_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A4 @ I3 ) @ ( B3 @ I3 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B3 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_3534_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q2: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( P @ X3 )
& ( Q2 @ X3 @ Y3 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X3: A] : ( collect @ B @ ( Q2 @ X3 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_3535_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A4: set @ B,F2: B > ( set @ A )] :
( ( ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3536_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I4: set @ A,J5: set @ A,C3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I4 @ J5 ) @ C3 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C3 ) @ ( product_Sigma @ A @ B @ J5 @ C3 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_3537_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I4: set @ A,J5: set @ A,C3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I4 @ J5 ) @ C3 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C3 ) @ ( product_Sigma @ A @ B @ J5 @ C3 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_3538_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C3: set @ A,A4: set @ B,B3: set @ B] :
( ( member @ A @ X @ C3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : C3 )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : C3 ) )
= ( ord_less_eq @ ( set @ B ) @ A4 @ B3 ) ) ) ).
% Times_subset_cancel2
thf(fact_3539_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I4: set @ A,X7: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I4 @ X7 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ I4 )
=> ( ( X7 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_3540_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ A,D4: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ ( product_Sigma @ A @ B @ C3
@ ^ [Uu: A] : D4 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ C3 )
@ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B3 @ D4 ) ) ) ).
% Times_Int_Times
thf(fact_3541_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ I3 ) @ ( B3 @ I3 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B3 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_3542_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ A,C3: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ B3 )
@ ^ [Uu: A] : C3 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C3 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : C3 ) ) ) ).
% Times_Int_distrib1
thf(fact_3543_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) ) ) ).
% finite_cartesian_product
thf(fact_3544_trancl__subset__Sigma,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ).
% trancl_subset_Sigma
thf(fact_3545_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A4 @ I3 ) @ ( B3 @ I3 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B3 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_3546_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ A,C3: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A4 @ B3 )
@ ^ [Uu: A] : C3 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C3 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : C3 ) ) ) ).
% Times_Un_distrib1
thf(fact_3547_Sigma__Union,axiom,
! [B: $tType,A: $tType,X7: set @ ( set @ A ),B3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X7 ) @ B3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A6: set @ A] : ( product_Sigma @ A @ B @ A6 @ B3 )
@ X7 ) ) ) ).
% Sigma_Union
thf(fact_3548_Id__on__subset__Times,axiom,
! [A: $tType,A4: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id_on @ A @ A4 )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ).
% Id_on_subset_Times
thf(fact_3549_rtrancl__last__touch,axiom,
! [A: $tType,Q4: A,Q6: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q6 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ A @ Q4 @ S )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_3550_rtrancl__last__visit_H,axiom,
! [A: $tType,Q4: A,Q6: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q6 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_3551_times__subset__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,C3: set @ B,B3: set @ A,D4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C3 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : D4 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( C3
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
& ( ord_less_eq @ ( set @ B ) @ C3 @ D4 ) ) ) ) ).
% times_subset_iff
thf(fact_3552_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_3553_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( B3 @ X3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( finite_finite2 @ B @ ( B3 @ A8 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) ) ) ) ).
% finite_SigmaI2
thf(fact_3554_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_3555_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B3 ) ) ) ).
% finite_cartesian_productD2
thf(fact_3556_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B3
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B3 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_3557_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W2: A,E3: set @ ( product_prod @ A @ A ),U4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ U4
@ ^ [Uu: A] : U4 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_rtrancl @ A @ E3 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_3558_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W2: A,E3: set @ ( product_prod @ A @ A ),U4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ U4
@ ^ [Uu: A] : U4 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E3 ) ) ) ).
% Restr_trancl_mono
thf(fact_3559_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E3: C > ( set @ A ),F5: D > ( set @ B ),A4: set @ C,B3: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A5: C,B4: D] :
( product_Sigma @ A @ B @ ( E3 @ A5 )
@ ^ [Uu: A] : ( F5 @ B4 ) ) )
@ ( product_Sigma @ C @ D @ A4
@ ^ [Uu: C] : B3 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E3 @ A4 ) )
@ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F5 @ B3 ) ) ) ) ).
% UN_Times_distrib
thf(fact_3560_homo__rel__restrict__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B3: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A4 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A4 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_3561_swap__product,axiom,
! [B: $tType,A: $tType,A4: set @ B,B3: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I3: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I3 ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : A4 ) ) ).
% swap_product
thf(fact_3562_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A6: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A6 )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_3563_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
= ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B3 ) ) ) ).
% card_cartesian_product
thf(fact_3564_sum_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > C > A,B3: set @ C,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ X3 ) @ B3 )
@ A4 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G2 )
@ ( product_Sigma @ B @ C @ A4
@ ^ [Uu: B] : B3 ) ) ) ) ).
% sum.cartesian_product
thf(fact_3565_prod_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > C > A,B3: set @ C,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G2 @ X3 ) @ B3 )
@ A4 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G2 )
@ ( product_Sigma @ B @ C @ A4
@ ^ [Uu: B] : B3 ) ) ) ) ).
% prod.cartesian_product
thf(fact_3566_rtrancl__last__visit,axiom,
! [A: $tType,Q4: A,Q6: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q6 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_3567_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : R ) ) ) )
=> ( ~ ( member @ A @ U @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E3 @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_3568_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G2: D > B,A4: set @ C,B3: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F2 @ X3 ) @ ( G2 @ Y3 ) ) )
@ ( product_Sigma @ C @ D @ A4
@ ^ [Uu: C] : B3 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A4 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G2 @ B3 ) ) ) ).
% image_paired_Times
thf(fact_3569_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A4: set @ A,R: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ R )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A4 @ R )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ A4 @ R ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_3570_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : A4 ) )
= ( finite_card @ B @ A4 ) ) ).
% card_cartesian_product_singleton
thf(fact_3571_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B3: B > ( set @ C ),G2: B > C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( finite_finite2 @ C @ ( B3 @ X2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ X3 ) @ ( B3 @ X3 ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G2 ) @ ( product_Sigma @ B @ C @ A4 @ B3 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_3572_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: B > ( set @ C ),G2: B > C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( finite_finite2 @ C @ ( B3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G2 @ X3 ) @ ( B3 @ X3 ) )
@ A4 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G2 ) @ ( product_Sigma @ B @ C @ A4 @ B3 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_3573_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_3574_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A6: set @ A,B5: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X3: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B5 @ X3 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_3575_product__fold,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X3: A,Z5: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
@ Z5
@ B3 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A4 ) ) ) ) ).
% product_fold
thf(fact_3576_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),M: A,X7: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( insert2 @ A @ M @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : X7 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ M ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_3577_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),X7: set @ A,M: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ X7
@ ^ [Uu: A] : ( insert2 @ A @ M @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_3578_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X3: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X3 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_3579_rtrancl__last__visit__node,axiom,
! [A: $tType,S3: A,S5: A,R: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ S5 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ( S3 != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
| ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ Sh ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit_node
thf(fact_3580_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ~ ( finite_finite2 @ B @ B3 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_3581_Restr__subset,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Restr_subset
thf(fact_3582_Gr__incl,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B,B3: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A4 @ F2 )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B3 ) ) ).
% Gr_incl
thf(fact_3583_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ) ).
% uminus_int_def
thf(fact_3584_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M2: nat,N2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_3585_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A4: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F2 ) )
=> ( ( F2 @ X )
= Fx ) ) ).
% GrD2
thf(fact_3586_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A4: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F2 ) )
=> ( member @ A @ X @ A4 ) ) ).
% GrD1
thf(fact_3587_less__eq,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M @ N ) ) ).
% less_eq
thf(fact_3588_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_3589_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As9: A > B] :
! [I3: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J3 ) @ R4 )
=> ( ord_less_eq @ B @ ( As9 @ I3 ) @ ( As9 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_3590_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_3591_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_3592_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_3593_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_3594_Gcd__remove0__nat,axiom,
! [M4: set @ nat] :
( ( finite_finite2 @ nat @ M4 )
=> ( ( gcd_Gcd @ nat @ M4 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M4 @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_3595_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q5: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q5 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_3596_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_3597_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( P4 @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert2 @ B @ I @ I4 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P4 @ I4 ) ) )
& ( ~ ( member @ B @ I @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert2 @ B @ I @ I4 ) )
= ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I4 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_3598_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3599_minus__rat__cancel,axiom,
! [A3: int,B2: int] :
( ( fract @ ( uminus_uminus @ int @ A3 ) @ ( uminus_uminus @ int @ B2 ) )
= ( fract @ A3 @ B2 ) ) ).
% minus_rat_cancel
thf(fact_3600_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_3601_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_3602_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_3603_minus__rat,axiom,
! [A3: int,B2: int] :
( ( uminus_uminus @ rat @ ( fract @ A3 @ B2 ) )
= ( fract @ ( uminus_uminus @ int @ A3 ) @ B2 ) ) ).
% minus_rat
thf(fact_3604_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_3605_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_3606_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_3607_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H3: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H3 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H3 @ F1 )
@ ^ [X3: A] : ( H3 @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_3608_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 @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_3609_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_3610_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_3611_Gcd__nat__eq__one,axiom,
! [N4: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N4 )
=> ( ( gcd_Gcd @ nat @ N4 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_3612_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A4 )
=> ( ( gcd_Gcd @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_3613_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,I4: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G2 @ X3 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ I4 ) ) ) ).
% prod.non_neutral'
thf(fact_3614_eq__rat_I2_J,axiom,
! [A3: int] :
( ( fract @ A3 @ ( zero_zero @ int ) )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% eq_rat(2)
thf(fact_3615_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
= ( semiring_1_of_nat @ rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_3616_quotient__of__eq,axiom,
! [A3: int,B2: int,P4: int,Q4: int] :
( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A3 @ B2 ) ) ) ).
% quotient_of_eq
thf(fact_3617_One__rat__def,axiom,
( ( one_one @ rat )
= ( fract @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% One_rat_def
thf(fact_3618_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ ( one_one @ int ) )
= ( ring_1_of_int @ rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_3619_normalize__eq,axiom,
! [A3: int,B2: int,P4: int,Q4: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A3 @ B2 ) ) ) ).
% normalize_eq
thf(fact_3620_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( gcd_Gcd @ A @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_3621_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_3622_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G2 @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G2 @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H3 @ I4 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_3623_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ B,F2: B > A,G2: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( dvd_dvd @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% Gcd_mono
thf(fact_3624_Zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% Zero_rat_def
thf(fact_3625_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_3626_rat__number__expand_I3_J,axiom,
( ( numeral_numeral @ rat )
= ( ^ [K4: num] : ( fract @ ( numeral_numeral @ int @ K4 ) @ ( one_one @ int ) ) ) ) ).
% rat_number_expand(3)
thf(fact_3627_rat__number__collapse_I3_J,axiom,
! [W2: num] :
( ( fract @ ( numeral_numeral @ int @ W2 ) @ ( one_one @ int ) )
= ( numeral_numeral @ rat @ W2 ) ) ).
% rat_number_collapse(3)
thf(fact_3628_quotient__of__Fract,axiom,
! [A3: int,B2: int] :
( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% quotient_of_Fract
thf(fact_3629_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K4: A] :
( ( member @ A @ K4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K4 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_3630_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_3631_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ T2 )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_3632_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,H3: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H3 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H3 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_3633_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,G2: B > A,H3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G2 @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ T2 )
= ( groups1962203154675924110t_prod @ B @ A @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_3634_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N2: A] :
( ( member @ A @ N2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_3635_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image2 @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_3636_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,G2: B > A,H3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G2 @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( H3 @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G2 @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G2 @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_3637_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I5: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I5 )
& ( ( P6 @ X3 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I5 )
& ( ( P6 @ X3 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_3638_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_3639_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_3640_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_3641_one__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ B2 @ A3 ) ) ) ).
% one_less_Fract_iff
thf(fact_3642_Fract__less__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A3 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_3643_rat__number__collapse_I5_J,axiom,
( ( fract @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ).
% rat_number_collapse(5)
thf(fact_3644_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 )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_3645_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( fract @ ( plus_plus @ int @ M @ N ) @ N )
= ( plus_plus @ rat @ ( fract @ M @ N ) @ ( one_one @ rat ) ) ) ) ).
% Fract_add_one
thf(fact_3646_one__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A3 ) ) ) ).
% one_le_Fract_iff
thf(fact_3647_Fract__le__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A3 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_3648_rat__number__collapse_I4_J,axiom,
! [W2: num] :
( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) ) ).
% rat_number_collapse(4)
thf(fact_3649_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
= ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% rat_number_expand(5)
thf(fact_3650_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_3651_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_3652_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_3653_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_3654_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_3655_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_3656_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_3657_Gcd__int__eq,axiom,
! [N4: set @ nat] :
( ( gcd_Gcd @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N4 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N4 ) ) ) ).
% Gcd_int_eq
thf(fact_3658_Gcd__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K5 ) )
= ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_abs_eq
thf(fact_3659_Gcd__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ nat
@ ( image2 @ int @ nat
@ ^ [K4: int] : ( nat2 @ ( abs_abs @ int @ K4 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_3660_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K7: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K7 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_3661_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_3662_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_3663_Gcd__eq__Max,axiom,
! [M4: set @ nat] :
( ( finite_finite2 @ nat @ M4 )
=> ( ( M4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( gcd_Gcd @ nat @ M4 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M2: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
@ M4 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_3664_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu2: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N2: nat] :
( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_3665_take__bit__num__simps_I7_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_3666_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X3: nat] :
( collect @ nat
@ ^ [N2: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_3667_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_3668_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3669_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_3670_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3671_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3672_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C2: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_3673_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max_insert
thf(fact_3674_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_3675_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ).
% Max_in
thf(fact_3676_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ A8 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_3677_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
=> ! [A15: A] :
( ( member @ A @ A15 @ A4 )
=> ( ord_less_eq @ A @ A15 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_3678_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ( member @ A @ M @ A4 )
& ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_3679_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_3680_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A4 )
= M )
= ( ( member @ A @ M @ A4 )
& ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_3681_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3682_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Max_Sup
thf(fact_3683_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= ( lattic643756798349783984er_Max @ A @ X7 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_3684_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_3685_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K2: nat,M3: nat] :
( X
!= ( product_Pair @ nat @ nat @ K2 @ M3 ) ) ).
% prod_decode_aux.cases
thf(fact_3686_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X4: set @ nat] :
( if @ nat
@ ( X4
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X4 ) ) ) ) ).
% Sup_nat_def
thf(fact_3687_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3688_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M4: set @ A,N4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M4 @ N4 )
=> ( ( M4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M4 ) @ ( lattic643756798349783984er_Max @ A @ N4 ) ) ) ) ) ) ).
% Max_mono
thf(fact_3689_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H3: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H3 @ ( ord_max @ A @ X2 @ Y2 ) )
= ( ord_max @ A @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H3 @ ( lattic643756798349783984er_Max @ A @ N4 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H3 @ N4 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_3690_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B3 ) @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max.subset
thf(fact_3691_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_3692_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member @ A @ ( ord_max @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Max.closed
thf(fact_3693_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_3694_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_3695_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F2 @ X3 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F2 @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_3696_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K4: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K4 @ N2 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_3697_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_3698_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_3699_sup__Some,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( sup_sup @ A @ X @ Y ) ) ) ) ).
% sup_Some
thf(fact_3700_inf__Some,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( inf_inf @ A @ X @ Y ) ) ) ) ).
% inf_Some
thf(fact_3701_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A4: set @ A,F2: A > B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X3: A] : ( some @ B @ ( F2 @ X3 ) )
@ A4 ) ) ) ) ) ).
% Some_SUP
thf(fact_3702_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M2: A > ( option @ B ),P2: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( ( M2 @ K4 )
= ( some @ B @ V2 ) )
& ( P2 @ ( product_Pair @ A @ B @ K4 @ V2 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_3703_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_3704_less__eq__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ $o @ A @ $true
@ ^ [Z5: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z5 ) @ Y3 )
@ X3 ) ) ) ) ).
% less_eq_option_def
thf(fact_3705_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X3: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y3: A] :
( case_option @ $o @ A @ $true
@ ^ [Z5: A] : ( ord_less @ A @ Z5 @ Y3 )
@ X3 ) ) ) ) ) ).
% less_option_def
thf(fact_3706_top__option__def,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ( top_top @ ( option @ A ) )
= ( some @ A @ ( top_top @ A ) ) ) ) ).
% top_option_def
thf(fact_3707_sup__option__def,axiom,
! [A: $tType] :
( ( sup @ A )
=> ( ( sup_sup @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ Y3
@ ^ [X9: A] :
( case_option @ ( option @ A ) @ A @ X3
@ ^ [Z5: A] : ( some @ A @ ( sup_sup @ A @ X9 @ Z5 ) )
@ Y3 )
@ X3 ) ) ) ) ).
% sup_option_def
thf(fact_3708_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A4 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ) ).
% Some_Sup
thf(fact_3709_Some__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( some @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( complete_Inf_Inf @ ( option @ A )
@ ( image2 @ B @ ( option @ A )
@ ^ [X3: B] : ( some @ A @ ( F2 @ X3 ) )
@ A4 ) ) ) ) ).
% Some_INF
thf(fact_3710_finite__range__updI,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),A3: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F2 @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_3711_take__bit__num__simps_I6_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_3712_surj__int__encode,axiom,
( ( image2 @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_int_encode
thf(fact_3713_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_3714_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_3715_singleton__None__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) )
= ( none @ A ) ) ) ).
% singleton_None_Sup
thf(fact_3716_empty__upd__none,axiom,
! [B: $tType,A: $tType,X: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( none @ B ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_3717_inf__None__2,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: option @ A] :
( ( inf_inf @ ( option @ A ) @ X @ ( none @ A ) )
= ( none @ A ) ) ) ).
% inf_None_2
thf(fact_3718_inf__None__1,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [Y: option @ A] :
( ( inf_inf @ ( option @ A ) @ ( none @ A ) @ Y )
= ( none @ A ) ) ) ).
% inf_None_1
thf(fact_3719_sup__None__1,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [Y: option @ A] :
( ( sup_sup @ ( option @ A ) @ ( none @ A ) @ Y )
= Y ) ) ).
% sup_None_1
thf(fact_3720_sup__None__2,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X: option @ A] :
( ( sup_sup @ ( option @ A ) @ X @ ( none @ A ) )
= X ) ) ).
% sup_None_2
thf(fact_3721_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X3: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_3722_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_3723_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_3724_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_3725_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_3726_case__optionE,axiom,
! [A: $tType,P: $o,Q2: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q2 @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y2: A] :
( ( X
= ( some @ A @ Y2 ) )
=> ~ ( Q2 @ Y2 ) ) ) ) ).
% case_optionE
thf(fact_3727_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_3728_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_3729_inf__option__def,axiom,
! [A: $tType] :
( ( inf @ A )
=> ( ( inf_inf @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Z5: A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Aa3: A] : ( some @ A @ ( inf_inf @ A @ Z5 @ Aa3 ) )
@ Y3 )
@ X3 ) ) ) ) ).
% inf_option_def
thf(fact_3730_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K4: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K4 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K4 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K4 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_3731_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A5: nat,X3: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P6: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X3 )
@ A5 )
@ ( product_Pair @ nat @ num @ N2 @ M2 ) ) ) ) ).
% take_bit_num_code
thf(fact_3732_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q2: A > $o,X: option @ A,R: B > $o,F2: B,G2: A > B] :
( ( case_option @ $o @ A @ P @ Q2 @ X )
=> ( ( P
=> ( R @ F2 ) )
=> ( ! [Q7: A] :
( ( Q2 @ Q7 )
=> ( R @ ( G2 @ Q7 ) ) )
=> ( R @ ( case_option @ B @ A @ F2 @ G2 @ X ) ) ) ) ) ).
% disjE_realizer2
thf(fact_3733_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_3734_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_3735_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X3: A,Y3: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_3736_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_3737_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H3: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H3 @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H3 @ F1 )
@ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
@ ^ [X3: num] : ( H3 @ ( F32 @ X3 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_3738_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V: A] :
( ( ( M @ K )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_3739_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_3740_Sup__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) )
= ( ^ [A6: set @ ( option @ A )] :
( if @ ( option @ A )
@ ( ( A6
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( A6
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
@ ( none @ A )
@ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A6 ) ) ) ) ) ) ) ).
% Sup_option_def
thf(fact_3741_these__not__empty__eq,axiom,
! [A: $tType,B3: set @ ( option @ A )] :
( ( ( these @ A @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B3
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B3
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_3742_these__empty__eq,axiom,
! [A: $tType,B3: set @ ( option @ A )] :
( ( ( these @ A @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B3
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B3
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_3743_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_3744_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_3745_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_3746_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_3747_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_3748_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_3749_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_3750_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_3751_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_3752_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_3753_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(7)
thf(fact_3754_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A5: A,B4: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A5 ) @ ( set_ord_lessThan @ A @ B4 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_3755_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_3756_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_3757_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert2 @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_3758_Some__image__these__eq,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
= ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A4 )
& ( X3
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_3759_and__not__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_3760_surj__int__decode,axiom,
( ( image2 @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ int ) ) ) ).
% surj_int_decode
thf(fact_3761_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_3762_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F2 )
=> ( ! [N3: nat] :
( ( ( F2 @ N3 )
= ( F2 @ ( suc @ N3 ) ) )
=> ( ( F2 @ ( suc @ N3 ) )
= ( F2 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N8: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ N9 @ N8 )
=> ! [M7: nat] :
( ( ord_less_eq @ nat @ M7 @ N8 )
=> ( ( ord_less @ nat @ M7 @ N9 )
=> ( ord_less @ A @ ( F2 @ M7 ) @ ( F2 @ N9 ) ) ) ) )
& ! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ( F2 @ N8 )
= ( F2 @ N9 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_3763_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= F2 ) ).
% restrict_map_UNIV
thf(fact_3764_restrict__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),A4: set @ A,B3: set @ A] :
( ( restrict_map @ A @ B @ ( restrict_map @ A @ B @ M @ A4 ) @ B3 )
= ( restrict_map @ A @ B @ M @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% restrict_restrict
thf(fact_3765_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 )
@ ^ [X3: A] : ( none @ C ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_3766_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D4: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X3: A] : ( none @ B )
@ D4 )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_3767_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_3768_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D4 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D4 )
= ( restrict_map @ A @ B @ M @ D4 ) ) ) ) ).
% restrict_fun_upd
thf(fact_3769_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D4: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_3770_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D4 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_3771_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_3772_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F: A > B] :
! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% mono_def
thf(fact_3773_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% monoI
thf(fact_3774_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ).
% monoE
thf(fact_3775_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ).
% monoD
thf(fact_3776_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_3777_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,M: A,N: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_max @ B @ ( F2 @ M ) @ ( F2 @ N ) )
= ( F2 @ ( ord_max @ A @ M @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_3778_option_Omap__ident,axiom,
! [A: $tType,T4: option @ A] :
( ( map_option @ A @ A
@ ^ [X3: A] : X3
@ T4 )
= T4 ) ).
% option.map_ident
thf(fact_3779_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_3780_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F2: A > B,A4: A,B3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A4 @ B3 ) ) @ ( inf_inf @ B @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) ) ) ) ).
% mono_inf
thf(fact_3781_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F2: A > B,A4: A,B3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% mono_sup
thf(fact_3782_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: A > ( option @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A4 ) ) )
=> ( member @ A @ K @ A4 ) ) ).
% graph_restrictD(1)
thf(fact_3783_Rings_Omono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% Rings.mono_mult
thf(fact_3784_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% mono_Sup
thf(fact_3785_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F2 @ ( A4 @ X3 ) )
@ I4 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I4 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_3786_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F2 @ ( A4 @ X3 ) )
@ I4 ) ) ) ) ) ).
% mono_INF
thf(fact_3787_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% mono_Inf
thf(fact_3788_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),X: B,S3: set @ B] :
( ( ( M @ X )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ ( insert2 @ B @ X @ S3 ) ) )
= ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ S3 ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_3789_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: A > ( option @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A4 ) ) )
=> ( ( M @ K )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_3790_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X3: B] : ( some @ A @ ( F @ X3 ) ) ) ) ) ).
% map_option_case
thf(fact_3791_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_3792_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),D4: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_3793_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_3794_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_3795_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_3796_and__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M3: num] :
( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ( ( ? [M3: num] :
( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_3797_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ A,F2: A > B > B,G2: C > A,R: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) ) @ S )
=> ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G2 ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_3798_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_3799_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X7: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X3: A] : ( principal @ B @ ( F2 @ X3 ) )
@ X7 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X7 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_3800_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_3801_sup__principal,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( filter @ A ) @ ( principal @ A @ A4 ) @ ( principal @ A @ B3 ) )
= ( principal @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% sup_principal
thf(fact_3802_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% group_add_class.minus_comp_minus
thf(fact_3803_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% boolean_algebra_class.minus_comp_minus
thf(fact_3804_inf__principal,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A4 ) @ ( principal @ A @ B3 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% inf_principal
thf(fact_3805_SUP__principal,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I3: B] : ( principal @ A @ ( A4 @ I3 ) )
@ I4 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) ) ).
% SUP_principal
thf(fact_3806_DEADID_Oin__rel,axiom,
! [B: $tType] :
( ( ^ [Y5: B,Z4: B] : Y5 = Z4 )
= ( ^ [A5: B,B4: B] :
? [Z5: B] :
( ( member @ B @ Z5 @ ( top_top @ ( set @ B ) ) )
& ( ( id @ B @ Z5 )
= A5 )
& ( ( id @ B @ Z5 )
= B4 ) ) ) ) ).
% DEADID.in_rel
thf(fact_3807_top__eq__principal__UNIV,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( principal @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% top_eq_principal_UNIV
thf(fact_3808_map__fun_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3 )
= ( id @ ( A > B ) ) ) ).
% map_fun.identity
thf(fact_3809_set_Oidentity,axiom,
! [A: $tType] :
( ( vimage @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( set @ A ) ) ) ).
% set.identity
thf(fact_3810_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_3811_nat__def,axiom,
( nat2
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% nat_def
thf(fact_3812_mono__Int,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A4: set @ A,B3: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) @ ( inf_inf @ ( set @ B ) @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) ) ) ).
% mono_Int
thf(fact_3813_mono__Un,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A4: set @ A,B3: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% mono_Un
thf(fact_3814_surj__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% surj_id
thf(fact_3815_less__int__def,axiom,
( ( ord_less @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% less_int_def
thf(fact_3816_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_3817_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_3818_principal__eq__bot__iff,axiom,
! [A: $tType,X7: set @ A] :
( ( ( principal @ A @ X7 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X7
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_3819_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M4: B > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M4
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M4 ) @ Rep )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_3820_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ A @ A @ Abs @ Rep )
= ( id @ A ) ) ) ).
% type_copy_Abs_o_Rep
thf(fact_3821_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ A @ B @ B @ Rep @ Abs )
= ( id @ B ) ) ) ).
% type_copy_Rep_o_Abs
thf(fact_3822_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ) ).
% numeral_and_num
thf(fact_3823_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_3824_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atMost @ A @ K4 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_3825_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atMost @ A @ K4 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_3826_finite__subsets__at__top__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite5375528669736107172at_top @ A @ A4 )
= ( principal @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_3827_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_3828_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A5: A,B4: A] : ( if @ A @ ( Less_eq2 @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% ord.min_def
thf(fact_3829_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_3830_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A4: set @ A] :
( ( finite5375528669736107172at_top @ A @ A4 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_3831_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_3832_uminus__integer__def,axiom,
( ( uminus_uminus @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% uminus_integer_def
thf(fact_3833_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X4: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y9: set @ A] :
( ( finite_finite2 @ A @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ X4 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
& ( ord_less_eq @ ( set @ A ) @ X4 @ A6 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_3834_coinduct3__mono__lemma,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [F2: A > ( set @ B ),X7: set @ B,B3: set @ B] :
( ( order_mono @ A @ ( set @ B ) @ F2 )
=> ( order_mono @ A @ ( set @ B )
@ ^ [X3: A] : ( sup_sup @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F2 @ X3 ) @ X7 ) @ B3 ) ) ) ) ).
% coinduct3_mono_lemma
thf(fact_3835_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_3836_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G2: C > B,F5: set @ ( A > ( set @ C ) )] :
( ( bNF_collect @ A @ B @ ( image2 @ ( A > ( set @ C ) ) @ ( A > ( set @ B ) ) @ ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G2 ) ) @ F5 ) )
= ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G2 ) @ ( bNF_collect @ A @ C @ F5 ) ) ) ).
% image_o_collect
thf(fact_3837_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_max @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_3838_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.right_idem
thf(fact_3839_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.left_idem
thf(fact_3840_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_3841_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_3842_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_3843_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_3844_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z2 @ X )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_3845_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_3846_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_3847_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_3848_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_3849_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_3850_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_3851_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_3852_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_3853_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_3854_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y ) )
= X ) ) ).
% max_min_same(1)
thf(fact_3855_min__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(1)
thf(fact_3856_min__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(2)
thf(fact_3857_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_3858_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_3859_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_3860_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_3861_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(3)
thf(fact_3862_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_3863_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_3864_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_3865_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_3866_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_3867_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_3868_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,M: A,N: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_min @ B @ ( F2 @ M ) @ ( F2 @ N ) )
= ( F2 @ ( ord_min @ A @ M @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_3869_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def_raw
thf(fact_3870_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_3871_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_3872_complete__linorder__inf__min,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% complete_linorder_inf_min
thf(fact_3873_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B2 @ C2 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_3874_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% max_min_distrib2
thf(fact_3875_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_3876_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% min_max_distrib2
thf(fact_3877_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C2 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.left_commute
thf(fact_3878_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( ord_min @ A @ B4 @ A5 ) ) ) ) ).
% min.commute
thf(fact_3879_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.assoc
thf(fact_3880_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_3881_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_3882_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_3883_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def
thf(fact_3884_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_3885_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_3886_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C2 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_3887_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_3888_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% min.orderI
thf(fact_3889_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_3890_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_3891_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( ord_min @ A @ A5 @ B4 ) ) ) ) ) ).
% min.order_iff
thf(fact_3892_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_3893_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_3894_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_min @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_3895_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_min @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% min.absorb_iff2
thf(fact_3896_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_3897_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_3898_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_3899_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
| ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% min_less_iff_disj
thf(fact_3900_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_3901_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( ord_min @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_3902_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_3903_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_3904_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_3905_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_3906_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_3907_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A6: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A6 )
& ( X3
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_3908_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_3909_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_3910_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_3911_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_3912_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_3913_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ M5 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_3914_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ N @ M5 ) )
@ M ) ) ).
% min_Suc1
thf(fact_3915_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F5: set @ ( C > ( set @ B ) ),G2: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F5 ) @ G2 )
= ( bNF_collect @ A @ B
@ ( image2 @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F @ G2 )
@ F5 ) ) ) ).
% collect_comp
thf(fact_3916_collect__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_collect @ B @ A )
= ( ^ [F7: set @ ( B > ( set @ A ) ),X3: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( B > ( set @ A ) ) @ ( set @ A )
@ ^ [F: B > ( set @ A )] : ( F @ X3 )
@ F7 ) ) ) ) ).
% collect_def
thf(fact_3917_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_3918_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( sup_sup @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_3919_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_3920_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_3921_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_3922_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_3923_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X )
=> ! [A15: A] :
( ( member @ A @ A15 @ A4 )
=> ( ord_less_eq @ A @ A15 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_3924_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ A8 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_3925_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_3926_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= ( lattic5882676163264333800up_fin @ A @ X7 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_3927_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_3928_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_3929_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H3: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H3 @ ( sup_sup @ A @ X2 @ Y2 ) )
= ( sup_sup @ A @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H3 @ ( lattic5882676163264333800up_fin @ A @ N4 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H3 @ N4 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_3930_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B3 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_3931_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_3932_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_3933_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_3934_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A4 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3935_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_3936_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_3937_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( inf_inf @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_3938_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_3939_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_3940_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_min @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_3941_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_3942_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_3943_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A3 )
= A3 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_3944_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_3945_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_3946_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C2: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_3947_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_3948_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min_insert
thf(fact_3949_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_3950_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_3951_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ).
% Min_in
thf(fact_3952_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_3953_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A4 )
= M )
= ( ( member @ A @ M @ A4 )
& ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ M @ X3 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_3954_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_3955_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ( member @ A @ M @ A4 )
& ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ M @ X3 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_3956_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A4 )
=> ( ord_less_eq @ A @ X @ A15 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_3957_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ X @ A8 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_3958_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_3959_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= ( lattic643756798350308766er_Min @ A @ X7 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_3960_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Min_Inf
thf(fact_3961_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_3962_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ X @ A8 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_3963_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A4 )
=> ( ord_less_eq @ A @ X @ A15 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_3964_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= ( lattic7752659483105999362nf_fin @ A @ X7 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_3965_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_3966_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_3967_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M4: set @ A,N4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M4 @ N4 )
=> ( ( M4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N4 ) @ ( lattic643756798350308766er_Min @ A @ M4 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_3968_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H3: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H3 @ ( ord_min @ A @ X2 @ Y2 ) )
= ( ord_min @ A @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H3 @ ( lattic643756798350308766er_Min @ A @ N4 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H3 @ N4 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_3969_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset
thf(fact_3970_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_3971_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member @ A @ ( ord_min @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Min.closed
thf(fact_3972_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_3973_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( lattic643756798350308766er_Min @ A @ B3 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_3974_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F2 @ X3 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F2 @ S ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_3975_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_3976_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H3: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H3 @ ( inf_inf @ A @ X2 @ Y2 ) )
= ( inf_inf @ A @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H3 @ ( lattic7752659483105999362nf_fin @ A @ N4 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H3 @ N4 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_3977_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_3978_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_3979_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_3980_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_3981_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A4 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3982_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_3983_Code__Numeral_Osub__def,axiom,
( code_sub
= ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
@ ^ [M2: num,N2: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_3984_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_3985_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_3986_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_3987_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_3988_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_3989_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_3990_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_3991_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X3: B,Z5: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
@ Z5
@ B3 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_3992_total__on__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R3 ) ).
% total_on_empty
thf(fact_3993_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A5: A,B4: A] : ( if @ A @ ( Less_eq2 @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% ord.max_def
thf(fact_3994_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_3995_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F2: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu: A] : F2 ) ).
% comp_fun_commute_const
thf(fact_3996_total__onI,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( X2 != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R3 ) ) ) ) )
=> ( total_on @ A @ A4 @ R3 ) ) ).
% total_onI
thf(fact_3997_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ A @ A )] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_3998_total__on__lex__prod,axiom,
! [A: $tType,B: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),B3: set @ B,S3: set @ ( product_prod @ B @ B )] :
( ( total_on @ A @ A4 @ R3 )
=> ( ( total_on @ B @ B3 @ S3 )
=> ( total_on @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ ( lex_prod @ A @ B @ R3 @ S3 ) ) ) ) ).
% total_on_lex_prod
thf(fact_3999_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X3: A,A11: set @ A] : ( if @ ( set @ A ) @ ( P @ X3 ) @ ( insert2 @ A @ X3 @ A11 ) @ A11 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_4000_total__lex__prod,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S3 )
=> ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R3 @ S3 ) ) ) ) ).
% total_lex_prod
thf(fact_4001_comp__fun__commute__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( finite4664212375090638736ute_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% comp_fun_commute_def'
thf(fact_4002_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X3: C,Y3: A,A6: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z5: B,A11: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Z5 ) @ A11 ) @ A11 ) )
@ A6
@ S ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_4003_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_4004_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_4005_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One2: A,Times: A > A > A,P6: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ One2
@ ( Times @ P6 @ ( power2 @ A @ One2 @ Times @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_4006_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: A > B > B,A4: set @ A,Z2: B,Y: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A4 @ Y )
=> ( ( member @ A @ A3 @ A4 )
=> ? [Y10: B] :
( ( Y
= ( F2 @ A3 @ Y10 ) )
& ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y10 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_4007_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F2: A > B > B,Z2: B,X: B] :
( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ X )
=> ( X = Z2 ) ) ).
% empty_fold_graphE
thf(fact_4008_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F2: A > B > B,Z2: B] : ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ Z2 ) ).
% fold_graph.emptyI
thf(fact_4009_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X3: A,Y3: B,A6: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X3 @ S ) @ ( insert2 @ B @ Y3 @ A6 ) @ A6 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_4010_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F2: A > B > B,Z2: B,A1: set @ A,A22: B] :
( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A1 @ A22 )
=> ( ( ( A1
= ( bot_bot @ ( set @ A ) ) )
=> ( A22 != Z2 ) )
=> ~ ! [X2: A,A10: set @ A] :
( ( A1
= ( insert2 @ A @ X2 @ A10 ) )
=> ! [Y2: B] :
( ( A22
= ( F2 @ X2 @ Y2 ) )
=> ( ~ ( member @ A @ X2 @ A10 )
=> ~ ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A10 @ Y2 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_4011_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F: A > B > B,Z5: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z5 ) )
| ? [X3: A,A6: set @ A,Y3: B] :
( ( A12
= ( insert2 @ A @ X3 @ A6 ) )
& ( A23
= ( F @ X3 @ Y3 ) )
& ~ ( member @ A @ X3 @ A6 )
& ( finite_fold_graph @ A @ B @ F @ Z5 @ A6 @ Y3 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_4012_Finite__Set_Ofold__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold @ A @ B )
= ( ^ [F: A > B > B,Z5: B,A6: set @ A] : ( if @ B @ ( finite_finite2 @ A @ A6 ) @ ( the @ B @ ( finite_fold_graph @ A @ B @ F @ Z5 @ A6 ) ) @ Z5 ) ) ) ).
% Finite_Set.fold_def
thf(fact_4013_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X3 ) @ A6 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_4014_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M3 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit0 @ M3 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit1 @ M3 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit0 @ M3 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit1 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_4015_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S )
=> ( ( relcomp @ A @ B @ C @ R @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X3: A,Y3: B,A6: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z5: C,A11: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Z5 ) @ A11 ) @ A11 ) )
@ A6
@ S ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_4016_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( bit1 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( bit0 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit1 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_4017_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) ) ) ).
% Field_insert
thf(fact_4018_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
thf(fact_4019_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
thf(fact_4020_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ C ),T2: set @ ( product_prod @ A @ C ),R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ C ) ) @ S @ T2 ) @ R )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ S @ R ) @ ( relcomp @ A @ C @ B @ T2 @ R ) ) ) ).
% relcomp_distrib2
thf(fact_4021_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B ),T2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ R @ ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ S @ T2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ R @ S ) @ ( relcomp @ A @ C @ B @ R @ T2 ) ) ) ).
% relcomp_distrib
thf(fact_4022_Field__square,axiom,
! [A: $tType,X: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X
@ ^ [Uu: A] : X ) )
= X ) ).
% Field_square
thf(fact_4023_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_4024_Field__Un,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) )
= ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R3 ) @ ( field2 @ A @ S3 ) ) ) ).
% Field_Un
thf(fact_4025_FieldI2,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ J @ ( field2 @ A @ R ) ) ) ).
% FieldI2
thf(fact_4026_FieldI1,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ I @ ( field2 @ A @ R ) ) ) ).
% FieldI1
thf(fact_4027_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R3: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R3 @ S3 ) )
=> ~ ! [B7: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B7 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B7 @ C2 ) @ S3 ) ) ) ).
% relcompEpair
thf(fact_4028_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R3: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R3 @ S3 ) )
=> ~ ! [X2: A,Y2: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X2 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y2 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z3 ) @ S3 ) ) ) ) ).
% relcompE
thf(fact_4029_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),C2: C,S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R3 @ S3 ) ) ) ) ).
% relcomp.relcompI
thf(fact_4030_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R3 @ S3 ) )
= ( ? [A5: A,B4: B,C5: C] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R3 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C5 ) @ S3 ) ) ) ) ).
% relcomp.simps
thf(fact_4031_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R3 @ S3 ) )
=> ~ ! [B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B7 ) @ R3 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B7 @ A22 ) @ S3 ) ) ) ).
% relcomp.cases
thf(fact_4032_trancl__unfold,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( relcomp @ A @ A @ A @ ( transitive_trancl @ A @ R4 ) @ R4 ) ) ) ) ).
% trancl_unfold
thf(fact_4033_union__comp__emptyR,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A ),C3: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A4 @ C3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A4 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B3 @ C3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_4034_union__comp__emptyL,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),C3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A4 @ C3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B3 @ C3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B3 ) @ C3 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_4035_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),A4: set @ A,B3: set @ B,S3: set @ ( product_prod @ B @ C ),C3: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S3
@ ( product_Sigma @ B @ C @ B3
@ ^ [Uu: B] : C3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R3 @ S3 )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : C3 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_4036_R__subset__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) ) ).
% R_subset_Field
thf(fact_4037_Restr__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu: A] : ( field2 @ A @ R3 ) ) )
= R3 ) ).
% Restr_Field
thf(fact_4038_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: set @ ( product_prod @ A @ C ),R3: D > ( set @ ( product_prod @ C @ B ) ),I4: set @ D] :
( ( relcomp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R3 @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I3: D] : ( relcomp @ A @ C @ B @ S3 @ ( R3 @ I3 ) )
@ I4 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_4039_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R3: D > ( set @ ( product_prod @ A @ C ) ),I4: set @ D,S3: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R3 @ I4 ) ) @ S3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I3: D] : ( relcomp @ A @ C @ B @ ( R3 @ I3 ) @ S3 )
@ I4 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_4040_trancl__Int__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ S3 ) @ R3 ) @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ S3 ) ) ) ).
% trancl_Int_subset
thf(fact_4041_rel__restrict__Int__empty,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R @ A4 )
= R ) ) ).
% rel_restrict_Int_empty
thf(fact_4042_Field__Restr__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ A4 ) ).
% Field_Restr_subset
thf(fact_4043_trancl__subset__Field2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu: A] : ( field2 @ A @ R3 ) ) ) ).
% trancl_subset_Field2
thf(fact_4044_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) )
= ( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_4045_Total__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Total_Restr
thf(fact_4046_total__on__imp__Total__Restr,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A4 @ R3 )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% total_on_imp_Total_Restr
thf(fact_4047_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_4048_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S ) @ ( insert2 @ ( 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
thf(fact_4049_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ A @ A )] :
! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R4 ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( X3 != Y3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).
% cofinal_def
thf(fact_4050_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S ) @ ( insert2 @ ( 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
thf(fact_4051_ntrancl__Suc,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N ) @ R )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R ) ) ) ).
% ntrancl_Suc
thf(fact_4052_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_4053_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
= ( A3 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_4054_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_4055_rtrancl__reflcl__absorb,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( id2 @ A ) )
= ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl_reflcl_absorb
thf(fact_4056_rtrancl__reflcl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
= ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl_reflcl
thf(fact_4057_trancl__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( transitive_rtrancl @ A @ R3 ) ) ).
% trancl_reflcl
thf(fact_4058_IdE,axiom,
! [A: $tType,P4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( id2 @ A ) )
=> ~ ! [X2: A] :
( P4
!= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ).
% IdE
thf(fact_4059_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
=> ( A3 = B2 ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_4060_max__ext__additive,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,R: set @ ( product_prod @ A @ A ),C3: set @ A,D4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B3 ) @ ( max_ext @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C3 @ D4 ) @ ( max_ext @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) @ ( sup_sup @ ( set @ A ) @ B3 @ D4 ) ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_ext_additive
thf(fact_4061_rtrancl__trancl__reflcl,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R4 ) @ ( id2 @ A ) ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_4062_rtrancl__unfold,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ R4 ) ) ) ) ).
% rtrancl_unfold
thf(fact_4063_reflcl__set__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_4064_rtrancl__Int__subset,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ S3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ S3 ) @ R3 ) @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ S3 ) ) ) ).
% rtrancl_Int_subset
thf(fact_4065_max__ext_Omax__extI,axiom,
! [A: $tType,X7: set @ A,Y4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y4 )
=> ( ( Y4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X7 @ Y4 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_4066_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 ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A1 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_4067_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 ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_4068_Total__subset__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) )
=> ( ( R3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A8: A] :
( R3
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A8 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_4069_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B1: A,B22: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B1 @ ( insert2 @ A @ B22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R4 ) )
& ( ( ( A12 = B1 )
& ( A23 = B22 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B1 @ B22 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B1 @ B22 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B1 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B1 @ B22 ) )
& ( A12 = B1 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B22 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_4070_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_4071_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R4: A > A > $o,X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_4072_Field__bsqr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) )
= ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu: A] : ( field2 @ A @ R3 ) ) ) ).
% Field_bsqr
thf(fact_4073_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_4074_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_4075_Preorder__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_preorder_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Preorder_Restr
thf(fact_4076_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat,S3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S3 ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S3 ) ) @ N )
= ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_4077_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X3: A,A6: set @ A] : ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_4078_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
thf(fact_4079_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_4080_range__enumerate,axiom,
! [S: set @ nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( ( image2 @ nat @ nat @ ( infini527867602293511546merate @ nat @ S ) @ ( top_top @ ( set @ nat ) ) )
= S ) ) ).
% range_enumerate
thf(fact_4081_Linear__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_679001287576687338der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Linear_order_Restr
thf(fact_4082_Linear__order__in__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_4083_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_4084_linear__order__on__Restr,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( order_679001287576687338der_on @ A @ A4 @ R3 )
=> ( order_679001287576687338der_on @ A @ ( inf_inf @ ( set @ A ) @ A4 @ ( order_above @ A @ R3 @ X ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_above @ A @ R3 @ X )
@ ^ [Uu: A] : ( order_above @ A @ R3 @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_4085_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S ) )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S2: A] :
( ( member @ A @ S2 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S2 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_4086_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S
@ ( insert2 @ A
@ ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_4087_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R3 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_4088_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_4089_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( ( wf @ A @ R3 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% wf_insert
thf(fact_4090_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q2: A > $o] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_4091_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_4092_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q2: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_4093_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_4094_wf__induct__rule,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R3 )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R3 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_4095_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [Q: set @ A] :
( ? [X3: A] : ( member @ A @ X3 @ Q )
=> ? [X3: A] :
( ( member @ A @ X3 @ Q )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 )
=> ~ ( member @ A @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_4096_wf__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ).
% wf_not_refl
thf(fact_4097_wf__not__sym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 ) ) ) ).
% wf_not_sym
thf(fact_4098_wf__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ).
% wf_irrefl
thf(fact_4099_wf__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R3 )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R3 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct
thf(fact_4100_wf__asym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 ) ) ) ).
% wf_asym
thf(fact_4101_wfUNIVI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [P3: A > $o,X2: A] :
( ! [Xa2: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R3 )
=> ( P3 @ Y2 ) )
=> ( P3 @ Xa2 ) )
=> ( P3 @ X2 ) )
=> ( wf @ A @ R3 ) ) ).
% wfUNIVI
thf(fact_4102_wfI__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Q3: set @ A] :
( ( member @ A @ X2 @ Q3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Q3 )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R )
=> ~ ( member @ A @ Y2 @ Q3 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_min
thf(fact_4103_wfE__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
( ( wf @ A @ R )
=> ( ( member @ A @ X @ Q2 )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q2 )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R )
=> ~ ( member @ A @ Y6 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_4104_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [P2: A > $o] :
( ! [X3: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 )
=> ( P2 @ Y3 ) )
=> ( P2 @ X3 ) )
=> ! [X4: A] : ( P2 @ X4 ) ) ) ) ).
% wf_def
thf(fact_4105_wf__Int2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) ) ) ).
% wf_Int2
thf(fact_4106_wf__Int1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R5 ) ) ) ).
% wf_Int1
thf(fact_4107_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q2: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A8: A] :
( ( P @ A8 )
=> ( ! [B13: A] :
( ( P @ B13 )
=> ( ord_less_eq @ A @ A8 @ B13 ) )
=> ( Q2 @ A8 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_4108_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q2: A > $o] :
( ( P @ A3 )
=> ( ! [A8: A] :
( ( P @ A8 )
=> ( ! [B13: A] :
( ( P @ B13 )
=> ( ord_less_eq @ A @ A8 @ B13 ) )
=> ( Q2 @ A8 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_4109_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_4110_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ X2 @ Y6 ) )
=> ( Q2 @ X2 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_4111_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_4112_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_4113_wfE__min_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Q2: set @ A] :
( ( wf @ A @ R )
=> ( ( Q2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q2 )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R )
=> ~ ( member @ A @ Y6 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_4114_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),F2: nat > A] :
( ( wf @ A @ R3 )
=> ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ K2 ) ) @ ( F2 @ K2 ) ) @ R3 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_4115_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
~ ? [F: nat > A] :
! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_4116_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% Least_le
thf(fact_4117_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_4118_wf__no__loop,axiom,
! [B: $tType,R: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R @ R )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R ) ) ).
% wf_no_loop
thf(fact_4119_wf__union__merge,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) )
= ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ R ) @ ( relcomp @ A @ A @ A @ S @ R ) ) @ S ) ) ) ).
% wf_union_merge
thf(fact_4120_Inf__nat__def,axiom,
( ( complete_Inf_Inf @ nat )
= ( ^ [X4: set @ nat] :
( ord_Least @ nat
@ ^ [N2: nat] : ( member @ nat @ N2 @ X4 ) ) ) ) ).
% Inf_nat_def
thf(fact_4121_wf,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% wf
thf(fact_4122_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F2: A > nat,G2: A > A] :
( ! [X2: A] :
( ( P @ X2 )
=> ( ord_less @ nat @ ( F2 @ ( G2 @ X2 ) ) @ ( F2 @ X2 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,X3: A] :
( ( P @ X3 )
& ( Y3
= ( G2 @ X3 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_4123_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_4124_wf__bounded__measure,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A8: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A8 ) @ R3 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B7 ) @ ( Ub @ A8 ) )
& ( ord_less_eq @ nat @ ( F2 @ B7 ) @ ( Ub @ A8 ) )
& ( ord_less @ nat @ ( F2 @ A8 ) @ ( F2 @ B7 ) ) ) )
=> ( wf @ A @ R3 ) ) ).
% wf_bounded_measure
thf(fact_4125_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_4126_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R3 )
=> ( ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) )
=> ( ( P @ K )
=> ? [X2: B] :
( ( P @ X2 )
& ! [Y6: B] :
( ( P @ Y6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X2 ) @ ( M @ Y6 ) ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_4127_wf__union__compatible,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( wf @ A @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ) ).
% wf_union_compatible
thf(fact_4128_wfI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : B3 ) )
=> ( ! [X2: A,P3: A > $o] :
( ! [Xa2: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R3 )
=> ( P3 @ Y2 ) )
=> ( P3 @ Xa2 ) )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ X2 @ B3 )
=> ( P3 @ X2 ) ) ) )
=> ( wf @ A @ R3 ) ) ) ).
% wfI
thf(fact_4129_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ? [X_12: A] : ( P @ X_12 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_4130_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A] :
( ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S ) ) ) ) ).
% enumerate_0
thf(fact_4131_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A6: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R4 ) )
& ( A6
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_4132_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F2: A > ( set @ B )] :
( ! [A8: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A8 ) @ R3 )
=> ( ( finite_finite2 @ B @ ( Ub @ A8 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B7 ) @ ( Ub @ A8 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ B7 ) @ ( Ub @ A8 ) )
& ( ord_less @ ( set @ B ) @ ( F2 @ A8 ) @ ( F2 @ B7 ) ) ) )
=> ( wf @ A @ R3 ) ) ).
% wf_bounded_set
thf(fact_4133_qc__wf__relto__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) @ R ) )
=> ( ( wf @ A @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ S ) @ ( relcomp @ A @ A @ A @ R @ ( transitive_rtrancl @ A @ S ) ) ) )
= ( wf @ A @ R ) ) ) ).
% qc_wf_relto_iff
thf(fact_4134_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_4135_wf__bounded__supset,axiom,
! [A: $tType,S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [Q9: set @ A,Q: set @ A] :
( ( ord_less @ ( set @ A ) @ Q @ Q9 )
& ( ord_less_eq @ ( set @ A ) @ Q9 @ S ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_4136_finite__subset__wf,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X4: set @ A,Y9: set @ A] :
( ( ord_less @ ( set @ A ) @ X4 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A4 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_4137_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S )
=> ( ord_less_eq @ A @ X5 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y3: B] : ( member @ B @ Y3 @ ( image2 @ A @ B @ F2 @ S ) ) )
= ( F2
@ ( ord_Least @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_4138_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S2: A] :
( ( member @ A @ S2 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S2 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_4139_reduction__pairI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ).
% reduction_pairI
thf(fact_4140_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R )
=> ( ! [F3: A > B,G4: A > B,X2: A,R6: B] :
( ! [Z8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z8 @ X2 ) @ R )
=> ( ( F3 @ Z8 )
= ( G4 @ Z8 ) ) )
=> ( ( P @ F3 @ X2 @ R6 )
= ( P @ G4 @ X2 @ R6 ) ) )
=> ( ! [X2: A,F3: A > B] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
=> ( P @ F3 @ Y6 @ ( F3 @ Y6 ) ) )
=> ? [X_12: B] : ( P @ F3 @ X2 @ X_12 ) )
=> ? [F3: A > B] :
! [X5: A] : ( P @ F3 @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_4141_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S8: set @ A,P2: A > $o] :
( ord_Least @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ S8 )
& ( P2 @ X3 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_4142_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S8: set @ A,P2: A > $o] :
( ord_Least @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ S8 )
& ( P2 @ X3 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_4143_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 )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ X2 @ Z2 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) @ ( chains2 @ A @ S ) ) ) ) ) ).
% chains_extend
thf(fact_4144_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_4145_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType] :
( ( boolea3799213064322606851m_diff @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One2: A,Xor2: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One2 )
& ( boolea5476839437570043046axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Xor2 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_def
thf(fact_4146_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > 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
thf(fact_4147_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( member @ A @ A3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ).
% img_fst
thf(fact_4148_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_4149_fst__image__times,axiom,
! [B: $tType,A: $tType,B3: set @ B,A4: set @ A] :
( ( ( B3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_4150_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_div_numeral
thf(fact_4151_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_4152_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_fst @ A @ B @ X ) )
=> ( P @ A3 ) ) ) ).
% fstE
thf(fact_4153_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_4154_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4155_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 )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_4156_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: A > A > A,Disj: A > A > A,Conj: A > A > A,Compl: A > A] :
( ! [X2: A,Y2: A] :
( ( Xor @ X2 @ Y2 )
= ( Disj @ ( Conj @ X2 @ ( Compl @ Y2 ) ) @ ( Conj @ ( Compl @ X2 ) @ Y2 ) ) )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
thf(fact_4157_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType] :
( ( boolea5476839437570043046axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Xor2: A > A > A] :
! [X3: A,Y3: A] :
( ( Xor2 @ X3 @ Y3 )
= ( Disj2 @ ( Conj2 @ X3 @ ( Compl2 @ Y3 ) ) @ ( Conj2 @ ( Compl2 @ X3 ) @ Y3 ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
thf(fact_4158_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C] :
( ( ^ [X3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X3 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A5: A,Uu: B] : ( F2 @ A5 ) ) ) ).
% fn_fst_conv
thf(fact_4159_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X23: B] : X12 ) ) ).
% fst_def
thf(fact_4160_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4161_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_4162_vimage__fst,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
= ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_4163_fst__image__mp,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B3: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) @ B3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A4 )
=> ( member @ A @ X @ B3 ) ) ) ).
% fst_image_mp
thf(fact_4164_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B3 ) )
= ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( B3 @ X3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_4165_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
thf(fact_4166_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E5 @ ( graph @ A @ B @ M ) )
& ( ( product_fst @ A @ B @ E5 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_4167_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_4168_bezw_Osimps,axiom,
( bezw
= ( ^ [X3: nat,Y3: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y3
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4169_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X7: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S ) @ X7 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z5: B,A11: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A11 )
@ A11 ) )
@ X7
@ S ) ) ) ).
% insert_relcomp_union_fold
thf(fact_4170_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_4171_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4172_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( member @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ).
% img_snd
thf(fact_4173_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_4174_snd__image__times,axiom,
! [B: $tType,A: $tType,A4: set @ B,B3: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B3 ) )
= B3 ) ) ) ).
% snd_image_times
thf(fact_4175_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_mod_numeral
thf(fact_4176_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_4177_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C] :
( ( ^ [X3: product_prod @ A @ B] : ( F2 @ ( product_snd @ A @ B @ X3 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu: A] : F2 ) ) ).
% fn_snd_conv
thf(fact_4178_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_4179_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_4180_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_snd @ A @ B @ X ) )
=> ( P @ B2 ) ) ) ).
% sndE
thf(fact_4181_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 )
@ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_4182_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_4183_surjective__pairing,axiom,
! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
( T4
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% surjective_pairing
thf(fact_4184_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4185_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4186_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P4: A,Q2: B > $o,Q4: B] :
( ( P @ P4 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4187_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4188_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set @ ( product_prod @ B @ A )] :
( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S ) )
=> ~ ! [X2: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_4189_reduction__pair__lemma,axiom,
! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( wf @ A @ S )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_4190_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 )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_4191_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 )
@ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_4192_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_4193_vimage__snd,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A4 ) ) ).
% vimage_snd
thf(fact_4194_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P4: product_prod @ B @ A,Q2: C > $o,F2: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P4 ) @ ( product_fst @ B @ A @ P4 ) )
=> ( ! [X2: B,Y2: A] :
( ( P @ Y2 @ X2 )
=> ( Q2 @ ( F2 @ X2 @ Y2 ) ) )
=> ( Q2 @ ( product_case_prod @ B @ A @ C @ F2 @ P4 ) ) ) ) ).
% exE_realizer
thf(fact_4195_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C5: A > B > C,P6: product_prod @ A @ B] : ( C5 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% case_prod_unfold
thf(fact_4196_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F: A > B > C,X3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% case_prod_beta'
thf(fact_4197_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F2: A > B > C,G2: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F2 @ ( G2 @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X3: D] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% split_comp_eq
thf(fact_4198_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A4: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_4199_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B3: set @ B] :
( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
=> ( ( member @ B @ ( product_snd @ A @ B @ X ) @ B3 )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_4200_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X3: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X3 )
= ( product_snd @ A @ A @ X3 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_4201_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_4202_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_4203_snd__image__mp,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ B @ A ),B3: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) @ B3 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 )
=> ( member @ A @ Y @ B3 ) ) ) ).
% snd_image_mp
thf(fact_4204_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) ) ) ) ).
% snd_fst_flip
thf(fact_4205_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) ) ) ) ).
% fst_snd_flip
thf(fact_4206_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F2: D > C > A,G2: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F2 @ G2 ) @ X )
= ( F2 @ ( G2 @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_4207_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_4208_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A4: set @ B,B3: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A4 @ B3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% snd_image_Sigma
thf(fact_4209_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
@ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) ) ) ).
% subset_fst_snd
thf(fact_4210_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > ( product_prod @ B @ C ),A4: set @ B,B3: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F2
@ ( product_Sigma @ B @ C @ A4
@ ^ [Uu: B] : B3 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 ) @ A4 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) @ B3 ) ) ) ).
% vimage_Times
thf(fact_4211_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_4212_range__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F2 @ ( top_top @ ( set @ C ) ) )
@ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F2 ) @ ( top_top @ ( set @ C ) ) )
@ ^ [Uu: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F2 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_4213_ID_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( bNF_id_bnf @ ( A > B > $o ) )
= ( ^ [R2: A > B > $o,A5: A,B4: B] :
? [Z5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ Z5
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( insert2 @ ( product_prod @ A @ B ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) ) )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > A ) @ ( product_fst @ A @ B ) @ Z5 )
= A5 )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > B ) @ ( product_snd @ A @ B ) @ Z5 )
= B4 ) ) ) ) ).
% ID.in_rel
thf(fact_4214_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 )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R ) @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z5: B,A11: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A11 )
@ A11 ) )
@ ( relcomp @ C @ A @ B @ R @ S )
@ S ) ) ) ).
% insert_relcomp_fold
thf(fact_4215_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_4216_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_4217_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_4218_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_4219_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_4220_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P6 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_4221_chains__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ ( set @ ( set @ A ) )
@ ^ [C7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C7 @ A6 )
& ( chain_subset @ A @ C7 ) ) ) ) ) ).
% chains_def
thf(fact_4222_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R: A > B > $o,A3: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R
@ A3
@ B2 )
= ( ? [Z5: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z5 )
= A3 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z5 )
= B2 ) ) ) ) ).
% fun.in_rel
thf(fact_4223_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_4224_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_4225_gcd__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_neg2
thf(fact_4226_gcd__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_neg1
thf(fact_4227_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_4228_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_neg1_int
thf(fact_4229_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_neg2_int
thf(fact_4230_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_4231_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_4232_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_4233_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_4234_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_4235_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% Gcd_2
thf(fact_4236_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ).
% dup.rsp
thf(fact_4237_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R @ R @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ R
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_4238_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ R )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_4239_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ R
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_4240_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ^ [M2: num,N2: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) )
@ ^ [M2: num,N2: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) ) ) ).
% sub.rsp
thf(fact_4241_uminus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( uminus_uminus @ int )
@ ( uminus_uminus @ int ) ) ).
% uminus_integer.rsp
thf(fact_4242_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G2: B > C,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sa
@ X
@ ( comp @ B @ C @ D @ G2 @ Y ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G2 @ Y3 ) )
@ X
@ Y ) ) ).
% fun.rel_map(2)
thf(fact_4243_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sb
@ ( comp @ A @ C @ D @ I @ X )
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% fun.rel_map(1)
thf(fact_4244_fun_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,X: D > A,Ya: D > A,Y: D > B,Xa: D > B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image2 @ D @ B @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R
@ X
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_4245_fun_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,R: A > B > $o,X: D > A,Y: D > B,Ra: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R
@ X
@ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image2 @ D @ B @ Y @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_4246_fun_Orel__refl__strong,axiom,
! [A: $tType,B: $tType,X: B > A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ B @ A @ X @ ( top_top @ ( set @ B ) ) ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_rel_fun @ B @ B @ A @ A
@ ^ [Y5: B,Z4: B] : Y5 = Z4
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_4247_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add_mult
thf(fact_4248_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_4249_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > B > $o,R22: C > D > $o,P: A > C > $o,Q2: B > D > $o,A3: product_prod @ A @ B,A4: set @ ( product_prod @ A @ B ),B2: product_prod @ C @ D,B3: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R1
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R22
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ P
@ Q2 )
=> ( ( member @ ( product_prod @ A @ B ) @ A3 @ A4 )
=> ( ( member @ ( product_prod @ C @ D ) @ B2 @ B3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R1 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B3 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R22 ) ) )
=> ( ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_fst @ C @ D @ B2 ) )
= ( Q2 @ ( product_snd @ A @ B @ A3 ) @ ( product_snd @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_4250_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_4251_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_4252_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_4253_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_4254_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_4255_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] :
( ( dvd_dvd @ int @ D5 @ M )
& ( dvd_dvd @ int @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_4256_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4257_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ R
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_4258_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ R
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_4259_plus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_4260_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_4261_Gcd__in,axiom,
! [A4: set @ nat] :
( ! [A8: nat,B7: nat] :
( ( member @ nat @ A8 @ A4 )
=> ( ( member @ nat @ B7 @ A4 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A8 @ B7 ) @ A4 ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A4 ) @ A4 ) ) ) ).
% Gcd_in
thf(fact_4262_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_4263_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.comm_monoid_axioms
thf(fact_4264_gcd__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.monoid_axioms
thf(fact_4265_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_4266_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_4267_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M )
& ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_4268_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_4269_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ pcr_rat )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) )
@ fract ) ).
% Fract.transfer
thf(fact_4270_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_4271_times__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_4272_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_4273_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_4274_positive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X3: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
@ positive ) ).
% positive.transfer
thf(fact_4275_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_4276_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( T2
= ( ^ [X3: A,Y3: B] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T2
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [X3: A] : X3
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_4277_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_4278_positive__minus,axiom,
! [X: rat] :
( ~ ( positive @ X )
=> ( ( X
!= ( zero_zero @ rat ) )
=> ( positive @ ( uminus_uminus @ rat @ X ) ) ) ) ).
% positive_minus
thf(fact_4279_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ pcr_int
@ ^ [N2: nat] : ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_4280_uminus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_4281_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_4282_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_4283_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_4284_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_4285_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_4286_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_4287_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_4288_positive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X3: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ).
% positive_def
thf(fact_4289_of__rat_Otransfer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ A @ A @ pcr_rat
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [X3: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X3 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X3 ) ) )
@ ( field_char_0_of_rat @ A ) ) ) ).
% of_rat.transfer
thf(fact_4290_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F2: C > A,G2: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apsnd @ D @ B @ C @ G2 @ X ) )
= ( product_Pair @ A @ B @ ( F2 @ ( product_fst @ C @ D @ X ) ) @ ( G2 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4291_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G2: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G2 @ X ) )
= ( product_Pair @ A @ B @ ( G2 @ ( product_fst @ D @ C @ X ) ) @ ( F2 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4292_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F2: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F2 @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_4293_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
thf(fact_4294_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
thf(fact_4295_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
thf(fact_4296_of__rat__neg__one,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% of_rat_neg_one
thf(fact_4297_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_less_of_rat_iff
thf(fact_4298_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_4299_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_le_of_rat_iff
thf(fact_4300_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_4301_of__rat__neg__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W2: num] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% of_rat_neg_numeral_eq
thf(fact_4302_of__rat__sum,axiom,
! [A: $tType,B: $tType] :
( ( field_char_0 @ A )
=> ! [F2: B > rat,A4: set @ B] :
( ( field_char_0_of_rat @ A @ ( groups7311177749621191930dd_sum @ B @ rat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( field_char_0_of_rat @ A @ ( F2 @ A5 ) )
@ A4 ) ) ) ).
% of_rat_sum
thf(fact_4303_of__rat__prod,axiom,
! [A: $tType,B: $tType] :
( ( field_char_0 @ A )
=> ! [F2: B > rat,A4: set @ B] :
( ( field_char_0_of_rat @ A @ ( groups7121269368397514597t_prod @ B @ rat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A5: B] : ( field_char_0_of_rat @ A @ ( F2 @ A5 ) )
@ A4 ) ) ) ).
% of_rat_prod
thf(fact_4304_of__rat__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ A3 ) )
= ( uminus_uminus @ A @ ( field_char_0_of_rat @ A @ A3 ) ) ) ) ).
% of_rat_minus
thf(fact_4305_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_mult
thf(fact_4306_of__rat__def,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ A @ A @ rep_Rat @ ( id @ A )
@ ^ [X3: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X3 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ).
% of_rat_def
thf(fact_4307_plus__rat__def,axiom,
( ( plus_plus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% plus_rat_def
thf(fact_4308_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_4309_times__rat__def,axiom,
( ( times_times @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% times_rat_def
thf(fact_4310_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ).
% uminus_rat_def
thf(fact_4311_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_4312_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X3: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa4 @ X3 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_4313_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_4314_Fract__def,axiom,
( fract
= ( map_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat ) @ ( id @ int ) @ ( map_fun @ int @ int @ ( product_prod @ int @ int ) @ rat @ ( id @ int ) @ abs_Rat )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) ) ) ) ).
% Fract_def
thf(fact_4315_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_4316_inverse__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_4317_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F2: C > A,P4: product_prod @ C @ B] :
( ( Q4
= ( product_apfst @ C @ A @ B @ F2 @ P4 ) )
=> ~ ! [X2: C,Y2: B] :
( ( P4
= ( product_Pair @ C @ B @ X2 @ Y2 ) )
=> ( Q4
!= ( product_Pair @ A @ B @ ( F2 @ X2 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_4318_times__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_4319_one__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% one_rat.rsp
thf(fact_4320_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% zero_rat.rsp
thf(fact_4321_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ratrel )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) ) ) ).
% Fract.rsp
thf(fact_4322_of__rat_Orsp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ A @ A @ ratrel
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [X3: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X3 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X3 ) ) )
@ ^ [X3: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X3 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ).
% of_rat.rsp
thf(fact_4323_ratrel__def,axiom,
( ratrel
= ( ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X3 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y3 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_4324_times__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( 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 )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% times_rat.rsp
thf(fact_4325_uminus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ).
% uminus_rat.rsp
thf(fact_4326_positive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X3: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
@ ^ [X3: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ).
% positive.rsp
thf(fact_4327_inverse__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_4328_plus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( 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 )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% plus_rat.rsp
thf(fact_4329_uminus__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_4330_cr__rat__def,axiom,
( cr_rat
= ( ^ [X3: product_prod @ int @ int,Y3: rat] :
( ( ratrel @ X3 @ X3 )
& ( ( abs_Rat @ X3 )
= Y3 ) ) ) ) ).
% cr_rat_def
thf(fact_4331_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P4: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P4 ) )
= ( ? [A5: B] :
( P4
= ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_4332_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P4: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P4 ) )
= ( ? [B4: B] :
( P4
= ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_4333_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( 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 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_4334_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V ) )
= ( ( plus_plus @ nat @ X @ V )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_4335_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_4336_uminus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ).
% uminus_int.rsp
thf(fact_4337_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_4338_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_4339_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% of_int.rsp
thf(fact_4340_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] :
( ( plus_plus @ nat @ X3 @ V2 )
= ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% intrel_def
thf(fact_4341_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_int.rsp
thf(fact_4342_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_4343_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( 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 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_4344_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( 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 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_4345_quotient__of__def,axiom,
( quotient_of
= ( ^ [X3: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X3
= ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_4346_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( image @ A @ B @ R @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X3: A,Y3: B,A6: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X3 @ S ) @ ( insert2 @ B @ Y3 @ A6 ) @ A6 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_4347_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K4: int] : ( bit_concat_bit @ N2 @ K4 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_4348_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% trancl_set_ntrancl
thf(fact_4349_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_4350_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
thf(fact_4351_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A3 @ B2 ) )
= ( ( algebr8660921524188924756oprime @ A @ C2 @ A3 )
& ( algebr8660921524188924756oprime @ A @ C2 @ B2 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_4352_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
& ( algebr8660921524188924756oprime @ A @ B2 @ C2 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_4353_coprime__minus__left__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_minus_left_iff
thf(fact_4354_coprime__minus__right__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_minus_right_iff
thf(fact_4355_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_4356_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_4357_Image__empty1,axiom,
! [B: $tType,A: $tType,X7: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X7 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_4358_Image__Id__on,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A4 ) @ B3 )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Image_Id_on
thf(fact_4359_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
thf(fact_4360_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
thf(fact_4361_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_4362_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_4363_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_4364_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_gcd
thf(fact_4365_normalize__stable,axiom,
! [Q4: int,P4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
=> ( ( algebr8660921524188924756oprime @ int @ P4 @ Q4 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_4366_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E3: set @ ( product_prod @ B @ A )] :
( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E3 )
= ( image @ B @ A @ E3 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_4367_prod__coprime__left,axiom,
! [B: $tType,A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( algebr8660921524188924756oprime @ A @ ( F2 @ I2 ) @ A3 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ A3 ) ) ) ).
% prod_coprime_left
thf(fact_4368_prod__coprime__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ B,A3: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( F2 @ I2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_coprime_right
thf(fact_4369_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A4 ) )
=> ~ ! [X2: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R3 )
=> ~ ( member @ B @ X2 @ A4 ) ) ) ).
% ImageE
thf(fact_4370_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A4 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_4371_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_4372_Image__Un,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B3: set @ B] :
( ( image @ B @ A @ R @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B3 ) ) ) ).
% Image_Un
thf(fact_4373_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_4374_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_4375_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_4376_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_add_one_left
thf(fact_4377_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_4378_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_diff_one_left
thf(fact_4379_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 @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% divides_mult
thf(fact_4380_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 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_4381_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 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_4382_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A3 )
=> ( ( dvd_dvd @ A @ C4 @ B2 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprimeI
thf(fact_4383_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B4: A] :
! [C5: A] :
( ( dvd_dvd @ A @ C5 @ A5 )
=> ( ( dvd_dvd @ A @ C5 @ B4 )
=> ( dvd_dvd @ A @ C5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_4384_not__coprimeE,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 ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_4385_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
thf(fact_4386_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
thf(fact_4387_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
thf(fact_4388_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
thf(fact_4389_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
thf(fact_4390_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
thf(fact_4391_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
thf(fact_4392_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B3 ) ) ) ).
% Image_Int_subset
thf(fact_4393_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q4: A,R: set @ ( product_prod @ A @ A ),Q02: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q02 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q02 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_4394_rtrancl__image__advance,axiom,
! [A: $tType,Q4: A,R: set @ ( product_prod @ A @ A ),Q02: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q02 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ R )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q02 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_4395_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A3: A,N: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N )
=> ( ( modulo_modulo @ A @ A3 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_4396_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A3: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A3 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N )
=> ( ( modulo_modulo @ A @ A3 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_4397_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_4398_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_4399_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_4400_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_4401_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B4: A] :
( ( gcd_gcd @ A @ A5 @ B4 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_4402_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_4403_Un__Image,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( image @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) @ A4 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ S @ A4 ) ) ) ).
% Un_Image
thf(fact_4404_quotient__of__coprime,axiom,
! [R3: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P4 @ Q4 ) ) ).
% quotient_of_coprime
thf(fact_4405_normalize__coprime,axiom,
! [R3: product_prod @ int @ int,P4: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P4 @ Q4 ) ) ).
% normalize_coprime
thf(fact_4406_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),B3: C > ( set @ B ),A4: set @ C] :
( ( image @ B @ A @ R3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( image @ B @ A @ R3 @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% Image_UN
thf(fact_4407_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_4408_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A16: A,B9: A] :
( ( A3
= ( times_times @ A @ A16 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B9 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A16 @ B9 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_4409_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A7: A,B6: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A7 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B6 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A7 @ B6 ) ) ) ) ) ).
% gcd_coprime
thf(fact_4410_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_4411_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A10 @ ( image @ A @ A @ R @ A10 ) )
=> ( A10
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_4412_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( image @ A @ A @ R @ A4 ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_4413_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_4414_coprime__common__divisor__int,axiom,
! [A3: int,B2: int,X: int] :
( ( algebr8660921524188924756oprime @ int @ A3 @ B2 )
=> ( ( dvd_dvd @ int @ X @ A3 )
=> ( ( dvd_dvd @ int @ X @ B2 )
=> ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_4415_trancl__image__by__rtrancl,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),Vi: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ Vi ) @ Vi )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_4416_Image__singleton,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B4 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_4417_Image__subset,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),A4: set @ A,B3: set @ B,C3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R3 @ C3 ) @ B3 ) ) ).
% Image_subset
thf(fact_4418_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),B3: C > ( set @ B ),A4: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( image @ B @ A @ R3 @ ( B3 @ X3 ) )
@ A4 ) ) ) ).
% Image_INT_subset
thf(fact_4419_rtrancl__apply__insert,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,S: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( insert2 @ A @ X @ S ) )
= ( insert2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( sup_sup @ ( set @ A ) @ S @ ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_4420_rtrancl__Image__in__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ V5 ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R ) @ V5 ) ) ).
% rtrancl_Image_in_Field
thf(fact_4421_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ R ) @ R )
=> ( ~ ( member @ A @ V @ R )
=> ~ ( ~ ( member @ A @ U @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E3 @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_4422_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E3 ) )
=> ( ~ ( member @ A @ X @ R )
=> ( ~ ( member @ A @ Y @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ R ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E3 @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_4423_subset__Image__Image__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ A4 ) @ ( image @ A @ A @ R3 @ B3 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_4424_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),B5: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B5 ) ) ) ) ).
% Image_eq_UN
thf(fact_4425_Sigma__Image,axiom,
! [A: $tType,B: $tType,A4: set @ B,B3: B > ( set @ A ),X7: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A4 @ B3 ) @ X7 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ B ) @ X7 @ A4 ) ) ) ) ).
% Sigma_Image
thf(fact_4426_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X7: C > ( set @ ( product_prod @ B @ A ) ),I4: set @ C,S: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X7 @ I4 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I3: C] : ( image @ B @ A @ ( X7 @ I3 ) @ S )
@ I4 ) ) ) ).
% UN_Image
thf(fact_4427_finite__reachable__advance,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),V0: A,V: A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_4428_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_4429_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_4430_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ S ) @ S )
=> ( ( member @ A @ U @ S )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ S
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_4431_subset__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_4432_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_4433_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R4 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A6 ) ) ) ) ).
% quotient_def
thf(fact_4434_set__product,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs )
@ ^ [Uu: A] : ( set2 @ B @ Ys ) ) ) ).
% set_product
thf(fact_4435_card__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A4 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_4436_quotient__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_4437_quotient__is__empty,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A4 @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_4438_quotient__is__empty2,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A4 @ R3 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_4439_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ X @ A3 )
=> ( ( dvd_dvd @ nat @ X @ B2 )
=> ( X
= ( one_one @ nat ) ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_4440_quotientE,axiom,
! [A: $tType,X7: set @ A,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ~ ! [X2: A] :
( ( X7
= ( image @ A @ A @ R3 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X2 @ A4 ) ) ) ).
% quotientE
thf(fact_4441_quotientI,axiom,
! [A: $tType,X: A,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A4 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A4 @ R3 ) ) ) ).
% quotientI
thf(fact_4442_finite__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4443_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_4444_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_4445_finite__equiv__class,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( finite_finite2 @ A @ X7 ) ) ) ) ).
% finite_equiv_class
thf(fact_4446_finite__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4447_finite__quotient,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R3 ) ) ) ) ).
% finite_quotient
thf(fact_4448_card__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A4 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_4449_singleton__quotient,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_4450_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_4451_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_4452_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs1 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( modulo_modulo @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ A ) @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs22 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% mergesort_by_rel_split_length
thf(fact_4453_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_4454_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_4455_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_4456_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ( set2 @ A @ Ys2 )
= X )
& ( distinct @ A @ Ys2 ) ) ) ) ).
% distinct_finite_set
thf(fact_4457_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_4458_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_4459_distinct__finite__subset,axiom,
! [A: $tType,X: set @ A] :
( ( finite_finite2 @ A @ X )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ X )
& ( distinct @ A @ Ys2 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_4460_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_4461_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_4462_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F2: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F2 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( set2 @ A @ ( F2 @ X3 ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_4463_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4464_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4465_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_4466_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert2 @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_4467_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4468_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ).
% mergesort_by_rel_split.simps(1)
thf(fact_4469_length__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_4470_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4471_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A1: list @ A,P: ( A > A > $o ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [R8: A > A > $o,Xs3: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs3 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) )
=> ( P @ R8 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_4472_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R2: A > A > $o,Xs2: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs2 @ ( merges9089515139780605204_merge @ A @ R2 @ ( mergesort_by_rel @ A @ R2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) @ ( mergesort_by_rel @ A @ R2 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel.simps
thf(fact_4473_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa )
= Y )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.elims
thf(fact_4474_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list @ A,I4: ( set @ A ) > B > $o,Sigma_0: B,F2: B > A > B] :
( ( distinct @ A @ S )
=> ( ( I4 @ ( set2 @ A @ S ) @ Sigma_0 )
=> ( ! [X2: A,It: set @ A,Sigma: B] :
( ( member @ A @ X2 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S ) )
=> ( ( I4 @ It @ Sigma )
=> ( I4 @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F2 @ Sigma @ X2 ) ) ) ) )
=> ( I4 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F2 @ Sigma_0 @ S ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_4475_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_4476_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I3: nat,X3: A] : ( suc @ I3 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_4477_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R8: A > A > $o,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs3 ) ) ).
% mergesort_by_rel.cases
thf(fact_4478_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F2: A > B > B,A3: A,B2: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( ( F2 @ A3
@ ( foldl @ B @ A
@ ^ [A5: B,B4: A] : ( F2 @ B4 @ A5 )
@ B2
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A5: B,B4: A] : ( F2 @ B4 @ A5 )
@ ( F2 @ A3 @ B2 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_4479_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > C > A,G2: A > B > C > B,A3: A,B2: B,Xs: list @ C] :
( ( product_fst @ A @ B
@ ( foldl @ ( product_prod @ A @ B ) @ C
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
@ ^ [A5: A,B4: B,X3: C] : ( product_Pair @ A @ B @ ( F2 @ A5 @ X3 ) @ ( G2 @ A5 @ B4 @ X3 ) ) )
@ ( product_Pair @ A @ B @ A3 @ B2 )
@ Xs ) )
= ( foldl @ A @ C @ F2 @ A3 @ Xs ) ) ).
% fst_foldl
thf(fact_4480_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Zs: list @ A] :
( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_4481_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 )
= ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_4482_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R @ Xs )
= ( merges9089515139780605204_merge @ A @ R @ ( mergesort_by_rel @ A @ R @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.psimps
thf(fact_4483_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) ) ) ) ) ).
% mergesort_by_rel.pelims
thf(fact_4484_foldl__length__aux,axiom,
! [A: $tType,A3: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I3: nat,X3: A] : ( suc @ I3 )
@ A3
@ L )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_4485_foldl__set,axiom,
! [A: $tType,L: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
= ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [X3: set @ A] : ( member @ ( set @ A ) @ X3 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_4486_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_4487_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R: A > A > $o,X1: A,X22: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs12: list @ A,Xs23: list @ A] : ( merges9089515139780605204_merge @ A @ R @ ( mergesort_by_rel @ A @ R @ Xs12 ) @ ( mergesort_by_rel @ A @ R @ Xs23 ) )
@ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X22 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_4488_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys2 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_4489_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),N: nat,M: nat] :
( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N3 ) ) @ ( nth @ A @ Xs @ N3 ) ) @ R ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M ) ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% nth_step_trancl
thf(fact_4490_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_4491_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_4492_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_4493_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_4494_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_4495_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_4496_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs13: list @ A,Xs24: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A,X2: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [Xs13: list @ A,Xs24: list @ A,X13: A,X24: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X24 @ Xs3 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_4497_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) )
=> ( ! [R8: A > A > $o,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_4498_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_4499_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F3: A > B,A8: A,As4: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_4500_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P3: A > $o,Yes: list @ A,No: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P3 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P3: A > $o,Yes: list @ A,No: list @ A,X2: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P3 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_4501_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P3: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P3: A > B > $o,A8: A,As4: list @ A,B7: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ ( cons @ B @ B7 @ Bs2 ) ) ) )
=> ( ! [P3: A > B > $o,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P3: A > B > $o,V3: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_4502_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( ! [Xs3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_4503_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
=> ( ! [V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X13: A,L1: list @ A,X24: A,L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X24 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_4504_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F3: A > B > C,A8: A,As4: list @ A,B7: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ ( cons @ B @ B7 @ Bs2 ) ) ) )
=> ( ! [A8: A > B > C,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A8 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A8: A > B > C,V3: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A8 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_4505_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ~ ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_4506_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F3: A > B,X2: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ( ! [F3: A > B,X2: A,Y2: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ~ ! [A8: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A8 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_4507_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P3: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( nil @ A ) ) )
=> ~ ! [P3: A > A > $o,X2: A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_4508_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P3: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( nil @ A ) ) )
=> ( ! [P3: A > A > $o,X2: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [P3: A > A > $o,X2: A,Y2: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_4509_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,X1: A,X22: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs1 ) @ ( cons @ A @ X22 @ Xs22 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_4510_list__decomp__1,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( one_one @ nat ) )
=> ? [A8: A] :
( L
= ( cons @ A @ A8 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_4511_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 )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) ) ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) ) ) )
=> ~ ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X13: A,X24: A,Xs3: list @ A] :
( ( Xa
= ( cons @ A @ X13 @ ( cons @ A @ X24 @ Xs3 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X24 @ Xs24 ) ) @ Xs3 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_4512_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,X: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs22 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_4513_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_4514_lists__length__Suc__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N2: A] : ( cons @ A @ N2 @ Xs2 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) )
@ ^ [Uu: list @ A] : A4 ) ) ) ).
% lists_length_Suc_eq
thf(fact_4515_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys3 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_4516_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_4517_set__Cons__sing__Nil,axiom,
! [A: $tType,A4: set @ A] :
( ( set_Cons @ A @ A4 @ ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
@ A4 ) ) ).
% set_Cons_sing_Nil
thf(fact_4518_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y2: A,Ys3: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys3 ) )
=> ( ( ( ( X @ X2 @ Y2 )
=> ( Y
= ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) )
& ( ~ ( X @ X2 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V3 @ Va ) )
=> ( ( Y
= ( cons @ A @ V3 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_4519_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F: nat > nat] :
( ( order_mono @ nat @ nat @ F )
& ( ( image2 @ nat @ nat @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ ( F @ I3 ) ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
= ( ( F @ I3 )
= ( F @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4520_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I3 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_4521_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X3: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_4522_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( ( M @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M @ X3 ) ) )
= ( insert2 @ B @ V @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_4523_ran__map__option,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,M: B > ( option @ C )] :
( ( ran @ B @ A
@ ^ [X3: B] : ( map_option @ C @ A @ F2 @ ( M @ X3 ) ) )
= ( image2 @ C @ A @ F2 @ ( ran @ B @ C @ M ) ) ) ).
% ran_map_option
thf(fact_4524_listrel__Cons,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert2 @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert2 @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_4525_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_4526_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_4527_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F: B > A,A5: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F @ ( nth @ B @ Xs2 @ N2 ) ) @ ( power_power @ A @ A5 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4528_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_4529_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A3: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_4530_listrel__Nil,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert2 @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_4531_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R3 ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_4532_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R3 ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_4533_listrel_ONil,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R3 ) ) ).
% listrel.Nil
thf(fact_4534_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_4535_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_4536_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R3: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrel.Cons
thf(fact_4537_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R3 ) )
=> ~ ! [Y2: B,Ys3: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_4538_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R3 ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_4539_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys2 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_4540_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X2: A,Y2: B,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Ys3: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_4541_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N2 ) @ ( nth @ B @ Ys @ N2 ) ) @ R3 ) ) ) ) ).
% listrel_iff_nth
thf(fact_4542_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_4543_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_4544_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_4545_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_4546_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_4547_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_4548_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_4549_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_4550_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_4551_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J3 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_4552_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_4553_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_4554_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: nat,J: nat,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X )
=> ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
=> ~ ! [K2: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
=> ( ( ord_less @ nat @ K2 @ J )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K2 ) @ X )
=> ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_4555_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Pivot: A,X: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F2 @ Pivot @ ( cons @ B @ X @ Xs ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F2 @ X ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F2 @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F2 @ Pivot @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_4556_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_4557_Un__set__drop__extend,axiom,
! [A: $tType,J: nat,L: list @ ( set @ A )] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
=> ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% Un_set_drop_extend
thf(fact_4558_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
thf(fact_4559_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( linord4507533701916653071of_set @ A @ A4 )
= ( nil @ A ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4560_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_4561_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu: A,Uv: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_4562_drop__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs )
@ ^ [M2: nat] : ( drop @ A @ M2 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_4563_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_4564_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_4565_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_4566_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Pivot: A] :
( ( linorder_part @ B @ A @ F2 @ Pivot @ ( nil @ B ) )
= ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% part_code(1)
thf(fact_4567_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4568_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P @ X5 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_4569_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys2: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4570_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_4571_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_4572_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_4573_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,V: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I @ V )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs )
@ ^ [J3: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J3 @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_4574_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A3: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_4575_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F: B > A,A5: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( times_times @ A @ A5 ) @ ( F @ ( nth @ B @ Xs2 @ N2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4576_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
& ( P @ X5 ) ) ) ) ).
% extract_SomeE
thf(fact_4577_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
& ( P @ X3 ) ) ) ) ).
% extract_Some_iff
thf(fact_4578_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_4579_surj__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_4580_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_4581_append_Omonoid__axioms,axiom,
! [A: $tType] : ( monoid @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) ) ).
% append.monoid_axioms
thf(fact_4582_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F2: A > B > B,G2: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X3: A] : ( compow @ ( B > B ) @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_4583_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,G2: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( finite4664212375090638736ute_on @ A @ B @ S
@ ^ [X3: A] : ( compow @ ( B > B ) @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_4584_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F2: A > nat,X: A] :
( ( compow @ ( A > A ) @ ( F2 @ X ) @ ( times_times @ A @ X ) )
= ( times_times @ A @ ( power_power @ A @ X @ ( F2 @ X ) ) ) ) ) ).
% funpow_times_power
thf(fact_4585_set__union__code,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( set2 @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% set_union_code
thf(fact_4586_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_4587_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P4 ) @ P4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4588_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F2: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ P4 @ ( F2 @ P4 ) )
=> ( ord_less_eq @ A @ P4 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_4589_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_4590_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4591_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4592_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q2: A > A] :
( ( order_mono @ A @ A @ Q2 )
=> ( order_mono @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q2 @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4593_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_4594_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_4595_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_4596_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_4597_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K4: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K4 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_4598_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_4599_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A3: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_4600_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_4601_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_4602_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( Y
!= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs3 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X2 )
@ Xa ) ) ) ) ) ) ).
% subset_eq_mset_impl.elims
thf(fact_4603_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl: set @ ( list @ A ),Kl2: list @ A] :
( collect @ A
@ ^ [K4: A] : ( member @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K4 @ ( nil @ A ) ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_4604_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F: B > A,Xs2: list @ B] :
( case_list @ ( list @ B ) @ B @ ( nil @ B )
@ ^ [X3: B] :
( case_list @ ( list @ B ) @ B @ Xs2
@ ^ [Y3: B] :
( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) @ Xs2 @ ( cons @ B @ Y3 @ ( cons @ B @ X3 @ ( nil @ B ) ) ) )
@ ^ [Ab: B,List: list @ B] :
( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list @ B )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( list @ B )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( append @ B @ ( linorder_sort_key @ B @ A @ F @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F @ Gts ) ) ) )
@ ( linorder_part @ B @ A @ F @ ( F @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs2 ) ) ) )
@ Xs2 ) ) ) ) ).
% sort_key_by_quicksort_code
thf(fact_4605_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X3: int] : X3
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_4606_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H3: B > C,F1: B,F22: A > ( list @ A ) > B,List2: list @ A] :
( ( H3 @ ( case_list @ B @ A @ F1 @ F22 @ List2 ) )
= ( case_list @ C @ A @ ( H3 @ F1 )
@ ^ [X12: A,X23: list @ A] : ( H3 @ ( F22 @ X12 @ X23 ) )
@ List2 ) ) ).
% list.case_distrib
thf(fact_4607_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C2: B,Xs: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X3: A] : C2
@ Xs )
= Xs ) ) ).
% sort_key_const
thf(fact_4608_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs ) ) ) ).
% sorted_sort
thf(fact_4609_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_4610_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_4611_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y3: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( X = Y3 ) @ ( cons @ A @ Y3 @ Xs2 ) @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_4612_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( subset_eq_mset_impl @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [X3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Ys ) ) ) ).
% subset_eq_mset_impl.simps(2)
thf(fact_4613_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 )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( Y
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs3 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X2 )
@ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
thf(fact_4614_sort__by__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs )
= ( append @ A
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ ( filter2 @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs ) )
@ ( append @ A
@ ( filter2 @ A
@ ^ [X3: A] :
( X3
= ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs )
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ ( filter2 @ A @ ( ord_less @ A @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs ) ) ) ) ) ) ).
% sort_by_quicksort
thf(fact_4615_sort__key__by__quicksort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F: B > A,Xs2: list @ B] :
( append @ B
@ ( linorder_sort_key @ B @ A @ F
@ ( filter2 @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F @ X3 ) @ ( F @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs2 ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X3: B] :
( ( F @ X3 )
= ( F @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs2 )
@ ( linorder_sort_key @ B @ A @ F
@ ( filter2 @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F @ X3 ) )
@ Xs2 ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_4616_filter__filter,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q2 @ Xs ) )
= ( filter2 @ A
@ ^ [X3: A] :
( ( Q2 @ X3 )
& ( P @ X3 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_4617_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% set_filter
thf(fact_4618_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,K: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y3: A] :
( ( F2 @ Y3 )
= K )
@ ( linorder_sort_key @ A @ B @ F2 @ Xs ) )
= ( filter2 @ A
@ ^ [Y3: A] :
( ( F2 @ Y3 )
= K )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_4619_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_4620_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X3: A] :
( filter2 @ A
@ ^ [Y3: A] : X3 != Y3 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_4621_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List2: list @ A] :
( ( List2
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu: A,Uv: list @ A] : $true
@ List2 ) ) ).
% list.disc_eq_case(2)
thf(fact_4622_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List2: list @ A] :
( ( List2
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu: A,Uv: list @ A] : $false
@ List2 ) ) ).
% list.disc_eq_case(1)
thf(fact_4623_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_4624_inter__set__filter,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_4625_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G2: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X3: A] :
( X3
= ( G2 @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_4626_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X3: A] : X3 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_4627_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] : ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4628_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4629_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] : ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4630_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4631_length__filter__conv__card,axiom,
! [A: $tType,P4: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4632_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_4633_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F: B > A,Pivot2: A,Xs2: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F @ X3 ) @ Pivot2 )
@ Xs2 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X3: B] :
( ( F @ X3 )
= Pivot2 )
@ Xs2 )
@ ( filter2 @ B
@ ^ [X3: B] : ( ord_less @ A @ Pivot2 @ ( F @ X3 ) )
@ Xs2 ) ) ) ) ) ) ).
% part_def
thf(fact_4634_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs ) @ P )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs ) @ P )
@ ^ [X3: A,Xs2: list @ A] : X3
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs ) ) ) ) ) ).
% Bleast_code
thf(fact_4635_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys3 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_4636_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_4637_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( P @ ( nil @ A ) @ Ys3 ) )
=> ( ! [Xs3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ( P @ Xs3 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
=> ( ( P @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_4638_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert2 @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_4639_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_4640_quicksort_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( Y
!= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X2 @ Y3 )
@ Xs3 ) )
@ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X2 ) @ Xs3 ) ) ) ) ) ) ) ) ) ).
% quicksort.elims
thf(fact_4641_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( linorder_quicksort @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X @ Y3 )
@ Xs ) )
@ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X ) @ Xs ) ) ) ) ) ) ).
% quicksort.simps(2)
thf(fact_4642_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( P @ ( nil @ A ) @ Ys3 ) )
=> ( ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) )
=> ( ( P @ Ys3 @ Xs3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_4643_sort__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( linorder_quicksort @ A ) ) ) ).
% sort_quicksort
thf(fact_4644_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( Y
= ( cons @ A @ X2 @ ( splice @ A @ Xa @ Xs3 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_4645_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ A @ A @ X
@ ^ [A5: A,List: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_4646_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs2: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X3: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X3 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys2: list @ B] :
( Ys2
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_4647_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_4648_foldr__filter,axiom,
! [A: $tType,B: $tType,F2: B > A > A,P: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( foldr @ B @ A
@ ^ [X3: B] : ( if @ ( A > A ) @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_4649_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F2: A > B > B,Xs: list @ A,A3: B] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( ( foldr @ A @ B @ F2 @ Xs @ A3 )
= ( foldl @ B @ A
@ ^ [A5: B,B4: A] : ( F2 @ B4 @ A5 )
@ A3
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_4650_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A3: nat] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ A3 )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_4651_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F: B > A,A5: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X3: B,B4: A] : ( plus_plus @ A @ ( F @ X3 ) @ ( times_times @ A @ A5 @ B4 ) )
@ Xs2
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4652_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_4653_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_4654_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X3: list @ A] :
( X3
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_4655_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
=> ( ! [Ys3: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_4656_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R3 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R3 ) )
| ( ( M = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_4657_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R3 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_4658_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_4659_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( concat @ A
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xs ) )
= ( concat @ A @ Xs ) ) ).
% concat_filter_neq_Nil
thf(fact_4660_lenlex__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R3 ) ) ) ).
% lenlex_irreflexive
thf(fact_4661_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R3 ) ) ).
% Nil_lenlex_iff2
thf(fact_4662_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X3: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P2 @ X3 ) @ ( cons @ A @ X3 @ Xt ) @ Xt )
@ Xs2
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_4663_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_4664_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 ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append1
thf(fact_4665_total__lenlex,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lenlex @ A @ R3 ) ) ) ).
% total_lenlex
thf(fact_4666_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,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4667_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_4668_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys2: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys2 ) )
& ! [Ys2: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_4669_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_4670_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_4671_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( Y
!= ( case_list @ A @ A @ X2
@ ^ [A5: A,List: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_4672_mod__h__bot__normalize,axiom,
! [A: $tType,H3: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H3 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_normalize
thf(fact_4673_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X23: A] : X23 ) ) ).
% option.the_def
thf(fact_4674_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_4675_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y != X2 ) )
=> ( ! [X2: A,Y2: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_4676_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y = X2 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X2: A,Y2: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_4677_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R3 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_4678_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R3 ) ) ).
% Nil2_notin_lex
thf(fact_4679_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R3 ) ) ).
% Nil_notin_lex
thf(fact_4680_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) ) ) ).
% lex_append_leftI
thf(fact_4681_lex__append__left__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_left_iff
thf(fact_4682_lex__append__leftD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_leftD
thf(fact_4683_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_rightI
thf(fact_4684_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_4685_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R4 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4686_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4687_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 ) )
= ( mergesort_remdups @ A ) ) ) ).
% remdup_sort_mergesort_remdups
thf(fact_4688_merge__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L12: list @ A,L23: list @ A] :
( ( ( distinct @ A @ L12 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L12 ) )
=> ( ( ( distinct @ A @ L23 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L23 ) )
=> ( ( distinct @ A @ ( merge @ A @ L12 @ L23 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge @ A @ L12 @ L23 ) )
& ( ( set2 @ A @ ( merge @ A @ L12 @ L23 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L12 ) @ ( set2 @ A @ L23 ) ) ) ) ) ) ) ).
% merge_correct
thf(fact_4689_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4690_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4691_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_4692_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 ) ) ) ).
% comp_fun_commute_insort
thf(fact_4693_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_4694_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( finite_fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_4695_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ A3
@ ( remove1 @ A @ A3 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_4696_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4697_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ A3
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_4698_merge_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( X
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( cons @ A @ V3 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X13: A,L1: list @ A] :
( ( X
= ( cons @ A @ X13 @ L1 ) )
=> ! [X24: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X24 @ L22 ) )
=> ( ( ( ( ord_less @ A @ X13 @ X24 )
=> ( Y
= ( cons @ A @ X13 @ ( merge @ A @ L1 @ ( cons @ A @ X24 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X13 @ X24 )
=> ( ( ( X13 = X24 )
=> ( Y
= ( cons @ A @ X13 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X13 != X24 )
=> ( Y
= ( cons @ A @ X24 @ ( merge @ A @ ( cons @ A @ X13 @ L1 ) @ L22 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X24 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_4699_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
= ( ? [Y3: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y3 ) @ R3 )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4700_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X3: A] : X3 ) ) ) ).
% sorted_list_of_set_def
thf(fact_4701_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_4702_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_4703_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I2
thf(fact_4704_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R3 ) ) ).
% not_listrel1_Nil
thf(fact_4705_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R3 ) ) ).
% not_Nil_listrel1
thf(fact_4706_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_4707_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_4708_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R3 ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% append_listrel1I
thf(fact_4709_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_4710_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I1
thf(fact_4711_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R3 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_4712_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [X2: A] :
( ( Xs
= ( cons @ A @ X2 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_4713_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_4714_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% listrel1I
thf(fact_4715_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ~ ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ! [Us2: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X2 @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us2 @ ( cons @ A @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_4716_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_4717_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_4718_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R4: A > A > $o,Xs2: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_4719_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs2: list @ A] : ( nth @ A @ Xs2 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_4720_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ).
% listrel_def
thf(fact_4721_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X3: A] : X3 )
= ( ^ [Xs2: list @ A] : Xs2 ) ) ).
% map_ident
thf(fact_4722_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X3: B] : ( cons @ A @ ( F2 @ X3 ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F2 @ Xs ) ) ).
% concat_map_singleton
thf(fact_4723_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X3: B,Y3: B] : ( R @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_4724_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G2: A > B > A,A3: A,F2: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G2 @ A3 @ ( map @ C @ B @ F2 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A5: A,X3: C] : ( G2 @ A5 @ ( F2 @ X3 ) )
@ A3
@ Xs ) ) ).
% foldl_map
thf(fact_4725_list_Omap__ident,axiom,
! [A: $tType,T4: list @ A] :
( ( map @ A @ A
@ ^ [X3: A] : X3
@ T4 )
= T4 ) ).
% list.map_ident
thf(fact_4726_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_4727_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X3: B,Y3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_4728_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X2: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss2 ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T3: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_4729_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T3: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_4730_map__by__foldl,axiom,
! [B: $tType,A: $tType,F2: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L2: list @ B,X3: A] : ( append @ B @ L2 @ ( cons @ B @ ( F2 @ X3 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F2 @ L ) ) ).
% map_by_foldl
thf(fact_4731_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,G2: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= ( G2 @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_4732_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: list @ A,Y3: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X3 @ Y3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_4733_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F: A > ( option @ B ),Xs2: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F )
@ ( filter2 @ A
@ ^ [X3: A] :
( ( F @ X3 )
!= ( none @ B ) )
@ Xs2 ) ) ) ) ).
% map_filter_def
thf(fact_4734_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_4735_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys2: list @ A] : ( nth @ A @ Ys2 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_4736_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 )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_4737_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 ) @ ( product_Pair @ B @ A @ K ) @ L ) )
= L ) ).
% map_snd_mk_fst
thf(fact_4738_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% sorted_wrt_map_linord
thf(fact_4739_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_4740_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y3 ) @ ( product_fst @ A @ B @ X3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% sorted_wrt_map_rev_linord
thf(fact_4741_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs2: list @ A,Ys2: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys2 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_4742_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X3: A,Y3: A] : ( P @ Y3 @ X3 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_4743_merge__list_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
( ( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L3: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( 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
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc22: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_4744_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F: A > B > A,A5: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X3: B,Y3: A] : ( F @ Y3 @ X3 )
@ ( rev @ B @ Xs2 )
@ A5 ) ) ) ).
% foldl_conv_foldr
thf(fact_4745_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F: B > A > A,Xs2: list @ B,A5: A] :
( foldl @ A @ B
@ ^ [X3: A,Y3: B] : ( F @ Y3 @ X3 )
@ A5
@ ( rev @ B @ Xs2 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_4746_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F2 @ Xs )
@ ^ [Y3: A] : ( cons @ A @ Y3 @ ( map_filter @ B @ A @ F2 @ Xs ) )
@ ( F2 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_4747_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_4748_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys2: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y3: A] : ( cons @ A @ Y3 @ Ys2 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_4749_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 ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_4750_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 ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
thf(fact_4751_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 )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_4752_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_4753_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 ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_4754_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_4755_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X3: B] : ( if @ ( option @ A ) @ ( P @ X3 ) @ ( some @ A @ ( F2 @ X3 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_4756_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X3: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_4757_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 ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_4758_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_4759_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A4 @ B3 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) ) ).
% member_product
thf(fact_4760_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A6: set @ A,B5: set @ B] :
( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B5 ) ) ) ).
% Product_Type.product_def
thf(fact_4761_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X3: A,Xs2: list @ A] :
( filter2 @ A
@ ^ [Y3: A] : Y3 != X3
@ ( rev @ A @ Xs2 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_4762_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_4763_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X3: list @ A] :
( X3
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_4764_length__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_4765_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X21: A,X222: list @ A] : X222 ) ) ).
% tl_def
thf(fact_4766_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z5: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_4767_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,T4: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X3: B] : ( ord_less @ A @ T4 @ ( F2 @ X3 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X3: B] : ( ord_less @ A @ T4 @ ( F2 @ X3 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_4768_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Ys2: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Ys2 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs2 ) ) ) ) ).
% extract_def
thf(fact_4769_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T3: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_4770_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss2 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T3: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_4771_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q2: B > $o,G2: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X3: B] :
( ( P @ X3 )
& ( Q2 @ X3 ) )
@ G2 )
= ( ^ [X3: A] :
( ( comp @ B @ $o @ A @ P @ G2 @ X3 )
& ( comp @ B @ $o @ A @ Q2 @ G2 @ X3 ) ) ) ) ).
% conj_comp_iff
thf(fact_4772_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y3: A] : Y3 = X
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_4773_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X3: A,Xa4: list @ A] : ( some @ A @ X3 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs2 ) ) ) ) ).
% find_dropWhile
thf(fact_4774_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X3: A] : X3 = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_4775_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X2: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss2 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_4776_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y3: A] : Y3 != X
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y3: A] : Y3 != X
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_4777_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y3: A] : Y3 != X
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y3: A] : Y3 != X
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_4778_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( Y
= ( case_list @ A @ A @ X2
@ ^ [A5: A,List: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X2 @ Xs3 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_4779_quicksort_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( Y
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X2 @ Y3 )
@ Xs3 ) )
@ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X2 ) @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ) ) ) ) ).
% quicksort.pelims
thf(fact_4780_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I3: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I3 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4781_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X3: nat] : X3
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_4782_map__add__upt_H,axiom,
! [Ofs: nat,A3: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
@ ( upt @ A3 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A3 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_4783_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_4784_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F2: nat > A,M: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F2 @ ( upt @ N @ M ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F2 @ K4 ) )
@ ( upt @ N @ M ) ) ) ).
% enumerate_map_upt
thf(fact_4785_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_4786_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I3: nat] : ( F2 @ ( suc @ I3 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_4787_upt__filter__extend,axiom,
! [U: nat,U5: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U5 )
=> ( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ U @ I2 )
& ( ord_less @ nat @ I2 @ U5 ) )
=> ~ ( P @ I2 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_4788_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_4789_map__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_4790_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K4: nat] : ( P @ ( nth @ A @ L @ K4 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_4791_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X2: nat,Y2: nat] :
( ( ord_less_eq @ nat @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_4792_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 @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) ) @ X @ ( none @ B ) ) ) ) ) ).
% map_of_distinct_upd4
thf(fact_4793_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 @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ Ys ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_4794_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4795_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_4796_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_4797_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 ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_4798_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 ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_4799_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 ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_4800_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: B > A,G2: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G2 @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_4801_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F2: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C2 @ ( F2 @ X3 ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4802_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_4803_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G2: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G2 @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_4804_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X2: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X2 ) ) ) ).
% weak_map_of_SomeI
thf(fact_4805_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_4806_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F2 ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_4807_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
= ( some @ C @ V ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
= ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_4808_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G2: A > B] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G2 @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_4809_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F2: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4810_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( groups8242544230860333062m_list @ A @ Xs )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X3: A] : X3
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_4811_finite__range__map__of,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A )] : ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_of @ B @ A @ Xys ) @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_range_map_of
thf(fact_4812_card__length__sum__list__rec,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N4 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N4 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N4 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4813_card__length__sum__list,axiom,
! [M: nat,N4: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N4 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N4 @ M ) @ ( one_one @ nat ) ) @ N4 ) ) ).
% card_length_sum_list
thf(fact_4814_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R3: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X3: C] : R3
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R3 ) ) ) ).
% sum_list_triv
thf(fact_4815_sum__list__Suc,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X3: A] : ( suc @ ( F2 @ X3 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_4816_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,Xs: list @ ( product_prod @ A @ C )] :
( ( map_of @ A @ B
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [K4: A,V2: C] : ( product_Pair @ A @ B @ K4 @ ( F2 @ V2 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_4817_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V ) )
=> ? [Ys3: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys3 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs2 ) ) )
& ( ( map_of @ B @ A @ Ys3 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_4818_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs2: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_4819_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V ) )
=> ( ~ ( P @ ( product_Pair @ B @ A @ K @ V ) )
=> ( ( distinct @ B @ ( map @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ Xs ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P @ Xs ) @ K )
= ( none @ A ) ) ) ) ) ).
% map_of_Some_filter_not_in
thf(fact_4820_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( map_of @ A @ B @ Xs @ K4 )
= ( some @ B @ V2 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_4821_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F2: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( F2 @ K4 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F2 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_4822_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 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_lookup
thf(fact_4823_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 @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd2
thf(fact_4824_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X7: set @ A,F2: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X7 )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( F2 @ X3 ) )
@ X7 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4825_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( F2 @ X3 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_4826_transpose__aux__filter__head,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T3: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_4827_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X21: A,X222: list @ A] : X21 ) ) ).
% hd_def
thf(fact_4828_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X3: A] :
( X3
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_4829_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_4830_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X3: B] :
( ( F2 @ A3 )
= ( F2 @ X3 ) )
@ Xs ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_4831_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F: A > $o,Xs2: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F @ Xs2 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F ) @ Xs2 ) ) ) ) ).
% partition_filter_conv
thf(fact_4832_partition__rev__filter__conv,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( rev @ A @ ( filter2 @ A @ P @ Xs ) ) @ Yes2 ) @ ( append @ A @ ( rev @ A @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) @ No2 ) ) ) ).
% partition_rev_filter_conv
thf(fact_4833_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs2: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_4834_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X: A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X @ Xs ) )
= ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) ) @ Xs ) ) ).
% partition_rev.simps(2)
thf(fact_4835_partition__rev_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) ) ).
% partition_rev.simps(1)
thf(fact_4836_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_4837_partition__P,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Yes2 ) )
=> ( P @ X5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ No2 ) )
=> ~ ( P @ X5 ) ) ) ) ).
% partition_P
thf(fact_4838_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes3: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_4839_partition__rev_Oelims,axiom,
! [A: $tType,X: 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
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X2: A,Xs3: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs3 ) )
=> ( Y
!= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs3 ) ) ) ) ) ) ).
% partition_rev.elims
thf(fact_4840_partition__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes2 ) @ ( set2 @ A @ No2 ) )
= ( set2 @ A @ Xs ) ) ) ).
% partition_set
thf(fact_4841_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y2 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( 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
thf(fact_4842_partition__rev_Opelims,axiom,
! [A: $tType,X: 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 )
=> ( ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X2: A,Xs3: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( Y
= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs3 ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% partition_rev.pelims
thf(fact_4843_merge__list_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y = L3 )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X ) @ Ls ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_4844_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_4845_empty__eq__map__add__iff,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),G2: A > ( option @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( map_add @ A @ B @ F2 @ G2 ) )
= ( ( F2
= ( ^ [X3: A] : ( none @ B ) ) )
& ( G2
= ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_4846_map__add__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B @ M
@ ^ [X3: A] : ( none @ B ) )
= M ) ).
% map_add_empty
thf(fact_4847_empty__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B
@ ^ [X3: A] : ( none @ B )
@ M )
= M ) ).
% empty_map_add
thf(fact_4848_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M12: A > ( option @ B ),M23: A > ( option @ B ),X3: A] : ( case_option @ ( option @ B ) @ B @ ( M12 @ X3 ) @ ( some @ B ) @ ( M23 @ X3 ) ) ) ) ).
% map_add_def
thf(fact_4849_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K4: A,V2: B,M2: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M2 @ K4 @ ( some @ B @ V2 ) ) )
@ Ps
@ M ) ) ).
% map_add_map_of_foldr
thf(fact_4850_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
= ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
@ ^ [Xs2: list @ ( product_prod @ A @ B ),F: A > ( option @ B )] : ( map_add @ A @ B @ F @ ( map_of @ A @ B @ Xs2 ) )
@ Xss
@ ^ [X3: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_4851_finite__range__map__of__map__add,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),L: list @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_add @ B @ A @ F2 @ ( map_of @ B @ A @ L ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_map_of_map_add
thf(fact_4852_merge__list_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ 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)
thf(fact_4853_merge__list_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= L ) ) ) ).
% merge_list.psimps(2)
thf(fact_4854_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) ).
% merge_list.psimps(3)
thf(fact_4855_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A ),L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) ) ).
% merge_list.psimps(4)
thf(fact_4856_merge__list_Opinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A0: list @ ( list @ A ),A1: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ A0 @ A1 ) )
=> ( ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L3: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A ),L3: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc22: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc22 ) @ Ls )
=> ( P @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_4857_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc23: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) ) )
=> ( ( merge_list @ A @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc23 ) @ Ls2 ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_4858_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_4859_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs3 ) )
=> ( Y
!= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( X @ Y3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs3 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_4860_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M: nat,A3: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_4861_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( tl @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% tl_replicate
thf(fact_4862_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_4863_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 ) @ ( product_Pair @ A @ B @ K ) @ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_fst_mk_fst
thf(fact_4864_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 )
@ ^ [X3: B] : ( product_Pair @ B @ A @ X3 @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_4865_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X3: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_4866_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_4867_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_4868_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I3: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_4869_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_4870_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_4871_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% sort_quicksort_by_rel
thf(fact_4872_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4873_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N3: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_4874_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_4875_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs3 ) )
=> ( ( Y
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( X @ Y3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs3 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_4876_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A1: list @ A,A22: list @ A,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ A0 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A1 @ A22 ) ) )
=> ( ! [R8: A > A > $o,Sl: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R8 @ Sl @ ( nil @ A ) ) )
=> ( ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs3: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs3 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs3 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R8 @ Sl @ Y6 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs3 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R8 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R8 @ Sl @ Y6 ) ) @ Xb2 ) ) )
=> ( P @ R8 @ Sl @ ( cons @ A @ X2 @ Xs3 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_4877_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
=> ( ( quicksort_by_rel @ A @ R @ Sl2 @ ( nil @ A ) )
= Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
thf(fact_4878_total__lexord,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% total_lexord
thf(fact_4879_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A4: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A4 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A4 ) ) ) ) ) ).
% nths_Cons
thf(fact_4880_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,T4: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T4 @ K )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K4: A] : ( product_Pair @ B @ C @ ( F2 @ K4 ) ) )
@ T4 )
@ ( F2 @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_4881_inj__on__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_4882_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ).
% inj_uminus
thf(fact_4883_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_4884_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_4885_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_4886_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
| ( ( A3 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_4887_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R3 ) )
= ( ? [A5: A,X3: list @ A] :
( Y
= ( cons @ A @ A5 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_4888_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B4: A] : ( divide_divide @ A @ B4 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_4889_inj__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_4890_inj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_4891_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_4892_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_4893_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C,A4: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F2 @ A4 ) ) ).
% inj_on_apfst
thf(fact_4894_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > C,A4: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A4 ) )
= ( inj_on @ B @ C @ F2 @ A4 ) ) ).
% inj_on_apsnd
thf(fact_4895_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ A3 @ A4 ) )
= ( ( inj_on @ A @ B @ F2 @ A4 )
& ~ ( member @ B @ ( F2 @ A3 ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4896_inj__mapD,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_4897_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F2 ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% fun.inj_map
thf(fact_4898_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F2 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_4899_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X3: A,Y3: C] : ( F2 @ X3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_4900_injD,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_4901_injI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ! [X2: A,Y2: A] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_4902_inj__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_4903_inj__def,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ! [X3: A,Y3: A] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inj_def
thf(fact_4904_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X3: A] : X3
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_4905_inj__on__id2,axiom,
! [A: $tType,A4: set @ A] :
( inj_on @ A @ A
@ ^ [X3: A] : X3
@ A4 ) ).
% inj_on_id2
thf(fact_4906_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A] :
( inj_on @ A @ A
@ ^ [B4: A] : ( plus_plus @ A @ B4 @ A3 )
@ A4 ) ) ).
% inj_on_add'
thf(fact_4907_inj__on__Int,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( ( inj_on @ A @ B @ F2 @ A4 )
| ( inj_on @ A @ B @ F2 @ B3 ) )
=> ( inj_on @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% inj_on_Int
thf(fact_4908_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: B > A,D4: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ B @ A @ F2 @ D4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D4 )
& ( member @ A @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_4909_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X7: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( F2 @ X3 ) )
@ X7 ) ).
% inj_on_convol_ident
thf(fact_4910_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( C2 @ X3 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_4911_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X3: A] : ( product_Pair @ B @ A @ ( C2 @ X3 ) @ X3 )
@ S ) ).
% inj_Pair(2)
thf(fact_4912_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A4: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A4 ) ) ) ).
% inj_on_mult
thf(fact_4913_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_4914_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_4915_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F2 @ A3 ) @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( member @ A @ A3 @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_4916_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F2 @ A4 )
= ( image2 @ A @ B @ F2 @ B3 ) )
= ( A4 = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_4917_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B2 @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X3: A] :
( ( B2
= ( F2 @ X3 ) )
& ! [Y3: A] :
( ( B2
= ( F2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_4918_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( ( ( image2 @ A @ B @ F2 @ A4 )
= ( image2 @ A @ B @ F2 @ B3 ) )
= ( A4 = B3 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_4919_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: C > A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) )
=> ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F2 @ G2 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% inj_compose
thf(fact_4920_map__injective,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_4921_inj__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_4922_inj__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).
% inj_of_nat
thf(fact_4923_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ A ),F2: A > B] :
( ( S
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A10: set @ A] :
( ( member @ ( set @ A ) @ A10 @ S )
=> ( inj_on @ A @ B @ F2 @ A10 ) )
=> ( inj_on @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_4924_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B4: A] : ( minus_minus @ A @ B4 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_4925_inj__singleton,axiom,
! [A: $tType,A4: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) ).
% inj_singleton
thf(fact_4926_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A5: B] : ( member @ A @ ( F2 @ A5 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_4927_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_4928_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,A4: set @ B] :
( ( inj_on @ B @ A @ G2 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X3: A] : X3
@ ( image2 @ B @ A @ G2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A4 ) ) ) ) ).
% sum.image_eq
thf(fact_4929_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,A4: set @ B] :
( ( inj_on @ B @ A @ G2 @ A4 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X3: A] : X3
@ ( image2 @ B @ A @ G2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A4 ) ) ) ) ).
% prod.image_eq
thf(fact_4930_inj__on__diff__nat,axiom,
! [N4: set @ nat,K: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N4 )
=> ( ord_less_eq @ nat @ K @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_4931_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F2: A > nat,A4: set @ A,G2: B > nat,B3: set @ B] :
( ( inj_on @ A @ nat @ F2 @ A4 )
=> ( ( inj_on @ B @ nat @ G2 @ B3 )
=> ( ! [A8: A,B7: B] :
( ( member @ A @ A8 @ A4 )
=> ( ( member @ B @ B7 @ B3 )
=> ( algebr8660921524188924756oprime @ nat @ ( F2 @ A8 ) @ ( G2 @ B7 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A5: A,B4: B] : ( times_times @ nat @ ( F2 @ A5 ) @ ( G2 @ B4 ) ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_4932_swap__inj__on,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I3: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I3 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_4933_lexord__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_irreflexive
thf(fact_4934_lexord__linear,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A8: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B7 ) @ R3 )
| ( A8 = B7 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A8 ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_linear
thf(fact_4935_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R3 ) ) ).
% lexord_Nil_right
thf(fact_4936_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V: list @ A,R3: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_leftI
thf(fact_4937_inj__split__Cons,axiom,
! [A: $tType,X7: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N2: A] : ( cons @ A @ N2 @ Xs2 ) )
@ X7 ) ).
% inj_split_Cons
thf(fact_4938_inj__of__char,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( inj_on @ char @ A @ ( comm_s6883823935334413003f_char @ A ) @ ( top_top @ ( set @ char ) ) ) ) ).
% inj_of_char
thf(fact_4939_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A4: set @ A,A17: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F: A > B] :
( ( inj_on @ A @ B @ F @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ A17 ) ) )
= ( ? [G: B > A] :
( ( image2 @ B @ A @ G @ A17 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4940_finite__UNIV__inj__surj,axiom,
! [A: $tType,F2: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4941_finite__UNIV__surj__inj,axiom,
! [A: $tType,F2: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4942_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_4943_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F2: A > B,C3: set @ A,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ( image2 @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B3 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_4944_image__Int,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B3 ) ) ) ) ).
% image_Int
thf(fact_4945_image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B3 ) ) ) ) ).
% image_set_diff
thf(fact_4946_map__inj__on,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_4947_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_4948_inj__vimage__image__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( vimage @ A @ B @ F2 @ ( image2 @ A @ B @ F2 @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_4949_finite__vimageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H3: B > A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ H3 @ F5 ) ) ) ) ).
% finite_vimageI
thf(fact_4950_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H3: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H3 @ A4 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H3 @ F5 ) @ A4 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_4951_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( remdups_adj @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( map @ A @ B @ F2 @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_4952_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( Y3
= ( F @ X3 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_4953_range__inj__infinite,axiom,
! [A: $tType,F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
=> ~ ( finite_finite2 @ A @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% range_inj_infinite
thf(fact_4954_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F2 @ X ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_4955_inj__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( inj_on @ nat @ A @ ( infini527867602293511546merate @ A @ S ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% inj_enumerate
thf(fact_4956_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N2: nat,Xs2: list @ A] : ( nths @ A @ Xs2 @ ( collect @ nat @ ( ord_less_eq @ nat @ N2 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_4957_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),F2: B > C] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ( member @ A @ J2 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A4 @ I2 ) @ ( A4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A4 @ J2 ) @ ( A4 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( inj_on @ B @ C @ F2 @ ( A4 @ I2 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_4958_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X3: A] :
( ( F2 @ Y )
= ( F2 @ X3 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_4959_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,F2: B > C,A4: A > ( set @ B )] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( inj_on @ B @ C @ F2 @ ( A4 @ I2 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_4960_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [N3: nat,F3: nat > A] :
( ( A4
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) )
& ( inj_on @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_4961_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ~ ! [F3: A > nat] :
( ? [N3: nat] :
( ( image2 @ A @ nat @ F3 @ A4 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
=> ~ ( inj_on @ A @ nat @ F3 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_4962_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [F3: A > nat,N3: nat] :
( ( ( image2 @ A @ nat @ F3 @ A4 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
& ( inj_on @ A @ nat @ F3 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_4963_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X2: A,Y2: A,Z3: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R3 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_4964_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R3 ) )
=> ( ! [A8: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A8 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_append_leftD
thf(fact_4965_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ? [B13: A,Z8: list @ A] :
( Y
= ( cons @ A @ B13 @ Z8 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_rightI
thf(fact_4966_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R3 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R3 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_4967_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_4968_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_4969_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R3 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_4970_infinite__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_4971_infinite__iff__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ( ~ ( finite_finite2 @ A @ S ) )
= ( ? [F: nat > A] :
( ( inj_on @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_4972_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,G2: A > B,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( inj_on @ A @ B @ G2 @ B3 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ G2 @ B3 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_4973_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,C3: set @ A,A4: set @ C,B3: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F2 @ C3 )
=> ( ! [X2: C] :
( ( member @ C @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( B3 @ X2 ) @ C3 ) )
=> ( ( member @ C @ J @ A4 )
=> ( ( image2 @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X3: C] : ( image2 @ A @ B @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_4974_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F2: A > A,S3: A] :
( ( ( compow @ ( A > A ) @ N @ F2 @ S3 )
= S3 )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( ord_less @ nat @ M3 @ N )
=> ( ( compow @ ( A > A ) @ M3 @ F2 @ S3 )
!= S3 ) ) )
=> ( inj_on @ nat @ A
@ ^ [K4: nat] : ( compow @ ( A > A ) @ K4 @ F2 @ S3 )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_4975_quotient__diff1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A5: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R3 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A4 @ R3 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) ) ) ) ) ).
% quotient_diff1
thf(fact_4976_nths__append,axiom,
! [A: $tType,L: list @ A,L4: list @ A,A4: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L4 ) @ A4 )
= ( append @ A @ ( nths @ A @ L @ A4 )
@ ( nths @ A @ L4
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A4 ) ) ) ) ) ).
% nths_append
thf(fact_4977_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S3: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X3: A] : ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ S3 ) ) )
@ Xs )
= ( nths @ A @ Xs @ S3 ) ) ) ).
% filter_in_nths
thf(fact_4978_length__nths,axiom,
! [A: $tType,Xs: list @ A,I4: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I4 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I4 ) ) ) ) ) ).
% length_nths
thf(fact_4979_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( inj_on @ A @ B @ F2 @ A4 )
& ( inj_on @ A @ B @ F2 @ B3 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_4980_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_4981_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F2 @ A4 ) )
= ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_4982_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_4983_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,D4: set @ A,A4: set @ B] :
( ( inj_on @ A @ B @ F2 @ D4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ D4 ) ) @ ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_4984_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R3: set @ ( product_prod @ A @ A ),V: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W2 @ Z2 ) ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_sufI
thf(fact_4985_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I4: set @ B] :
? [F3: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) @ ( product_Sigma @ B @ A @ I4 @ A4 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_4986_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
( nths @ A @ Xs2
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P2 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_4987_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_4988_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_4989_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A3: B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A4 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( F2 @ X3 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_4990_card__quotient__disjoint,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A4 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R3 ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_4991_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R4: A > A > $o,Xs2: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_4992_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I3 @ X )
= ( take @ A @ I3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R3 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_4993_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C3: set @ A,G2: A > B,X: A] :
( ( member @ A @ I @ C3 )
=> ( ( inj_on @ A @ B @ G2 @ C3 )
=> ( ( comp @ B @ A @ A
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G2 @ C3 ) ) @ ( the_inv_into @ A @ B @ C3 @ G2 @ I3 ) @ X )
@ G2
@ I )
= ( id @ A @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_4994_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_4995_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A6: set @ A,F: A > B,X3: B] :
( the @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( ( F @ Y3 )
= X3 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_4996_tl__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_4997_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_4998_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M: nat,L: list @ A,N: nat] :
( ( filter2 @ nat
@ ^ [I3: nat] : ( P @ ( nth @ A @ ( take @ A @ M @ L ) @ I3 ) )
@ ( upt @ N @ M ) )
= ( filter2 @ nat
@ ^ [I3: nat] : ( P @ ( nth @ A @ L @ I3 ) )
@ ( upt @ N @ M ) ) ) ).
% filter_upt_take_conv
thf(fact_4999_take__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M2: nat] : ( cons @ A @ X @ ( take @ A @ M2 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_5000_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_5001_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_5002_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( 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
thf(fact_5003_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R3 ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I2 @ Xs )
= ( take @ A @ I2 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys @ I2 ) ) @ R3 ) ) ) ) ) ).
% lex_take_index
thf(fact_5004_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F2: A > B,Z2: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 @ Z2 )
= ( id @ A @ Z2 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_5005_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G2: A > B,C3: set @ A,B3: set @ A,X: A] :
( ( inj_on @ A @ B @ G2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ B3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G2 @ C3 ) ) @ ( the_inv_into @ A @ B @ C3 @ G2 @ I3 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_5006_ex__inj,axiom,
! [A: $tType] :
( ( countable @ A )
=> ? [To_nat: A > nat] : ( inj_on @ A @ nat @ To_nat @ ( top_top @ ( set @ A ) ) ) ) ).
% ex_inj
thf(fact_5007_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X: B,Y: A,Z2: A] :
( ( ( M @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_5008_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( ( dom @ A @ B @ F2 )
= ( bot_bot @ ( set @ A ) ) )
= ( F2
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_5009_dom__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),N: A > ( option @ B )] :
( ( dom @ A @ B @ ( map_add @ A @ B @ M @ N ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ N ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_add
thf(fact_5010_dom__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),A4: set @ A] :
( ( dom @ A @ B @ ( restrict_map @ A @ B @ M @ A4 ) )
= ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M ) @ A4 ) ) ).
% dom_restrict
thf(fact_5011_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_5012_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M @ X3 ) ) )
= ( insert2 @ A @ K @ ( dom @ A @ B @ M ) ) ) ).
% map_update_eta_repair(1)
thf(fact_5013_dom__const_H,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( dom @ A @ B
@ ^ [X3: A] : ( some @ B @ ( F2 @ X3 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const'
thf(fact_5014_restrict__map__inv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_inv
thf(fact_5015_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F2: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ Y ) )
= ( insert2 @ A @ X @ ( dom @ A @ B @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_5016_ran__map__add,axiom,
! [B: $tType,A: $tType,M13: A > ( option @ B ),M24: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M13 ) @ ( dom @ A @ B @ M24 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M13 @ M24 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M13 ) @ ( ran @ A @ B @ M24 ) ) ) ) ).
% ran_map_add
thf(fact_5017_ran__add,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),G2: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( dom @ A @ B @ G2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F2 @ G2 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F2 ) @ ( ran @ A @ B @ G2 ) ) ) ) ).
% ran_add
thf(fact_5018_Func__is__emp,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A4 @ B3 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_5019_Func__non__emp,axiom,
! [A: $tType,B: $tType,B3: set @ A,A4: set @ B] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_5020_dom__map__option,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C > B,M: A > ( option @ C )] :
( ( dom @ A @ B
@ ^ [K4: A] : ( map_option @ C @ B @ ( F2 @ K4 ) @ ( M @ K4 ) ) )
= ( dom @ A @ C @ M ) ) ).
% dom_map_option
thf(fact_5021_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ A
@ ^ [A5: A] :
( ( M2 @ A5 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_5022_dom__if,axiom,
! [B: $tType,A: $tType,P: A > $o,F2: A > ( option @ B ),G2: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( collect @ A @ P ) )
@ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ G2 )
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) ) ) ).
% dom_if
thf(fact_5023_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F2 ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X2: A] :
( ( F2 @ X2 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_5024_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A4: A > ( option @ B ),B3: A > ( option @ B ),C3: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A4 ) @ ( dom @ A @ B @ B3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A4 @ ( map_add @ A @ B @ B3 @ C3 ) )
= ( map_add @ A @ B @ B3 @ ( map_add @ A @ B @ A4 @ C3 ) ) ) ) ).
% map_add_left_comm
thf(fact_5025_map__add__comm,axiom,
! [B: $tType,A: $tType,M13: A > ( option @ B ),M24: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M13 ) @ ( dom @ A @ B @ M24 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M13 @ M24 )
= ( map_add @ A @ B @ M24 @ M13 ) ) ) ).
% map_add_comm
thf(fact_5026_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B ),N: A > ( option @ B ),N10: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N10 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M8 ) @ ( dom @ A @ B @ N10 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N ) @ ( map_add @ A @ B @ M8 @ N10 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_5027_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A4: set @ B,K: B] :
( ( ( restrict_map @ B @ A @ M @ A4 @ K )
= ( none @ A ) )
= ( ~ ( member @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M ) @ A4 ) ) ) ) ).
% restrict_map_eq(1)
thf(fact_5028_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M2: A > ( option @ B )] :
( ( ( dom @ A @ B @ M2 )
= A4 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M2 ) @ B3 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_5029_rat__denum,axiom,
? [F3: nat > rat] :
( ( image2 @ nat @ rat @ F3 @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ rat ) ) ) ).
% rat_denum
thf(fact_5030_Func__empty,axiom,
! [B: $tType,A: $tType,B3: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( insert2 @ ( A > B )
@ ^ [X3: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_5031_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ( P
@ ^ [X3: A] : ( none @ B ) )
=> ( ! [K2: A,V3: B,M3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M3 @ K2 @ ( some @ B @ V3 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_5032_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( the2 @ B @ ( M2 @ X3 ) ) )
@ ( dom @ A @ B @ M2 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_5033_graph__map__add,axiom,
! [B: $tType,A: $tType,M13: A > ( option @ B ),M24: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M13 ) @ ( dom @ A @ B @ M24 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M13 @ M24 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M13 ) @ ( graph @ A @ B @ M24 ) ) ) ) ).
% graph_map_add
thf(fact_5034_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F2 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V2: B] :
( F2
= ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( some @ B @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_5035_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( the2 @ B @ ( M @ K4 ) ) )
@ Xs ) )
= M ) ) ).
% map_of_map_keys
thf(fact_5036_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A18: set @ B,B14: set @ A,F22: C > D,B23: set @ C,A25: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A18 )
= B14 )
=> ( ( inj_on @ C @ D @ F22 @ B23 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B23 ) @ A25 )
=> ( ( ( B23
= ( bot_bot @ ( set @ C ) ) )
=> ( A25
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B23 @ B14 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B23 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A25 @ A18 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_5037_dom__override__on,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),G2: A > ( option @ B ),A4: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F2 @ G2 @ A4 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( minus_minus @ ( set @ A ) @ A4 @ ( dom @ A @ B @ G2 ) ) ) ) )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( inf_inf @ ( set @ A ) @ A4 @ ( dom @ A @ B @ G2 ) ) ) ) ) ) ).
% dom_override_on
thf(fact_5038_surj__from__nat,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( image2 @ nat @ A @ ( from_nat @ A ) @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_from_nat
thf(fact_5039_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ( override_on @ A @ B @ F2 @ G2 @ ( bot_bot @ ( set @ A ) ) )
= F2 ) ).
% override_on_emptyset
thf(fact_5040_surj__nat__to__rat__surj,axiom,
( ( image2 @ nat @ rat @ nat_to_rat_surj @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ rat ) ) ) ).
% surj_nat_to_rat_surj
thf(fact_5041_dom__map__upds,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Xs: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_upds
thf(fact_5042_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 )
=> ( ( ( set_to_map @ A @ B @ S @ K )
= ( some @ B @ V ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ S ) ) ) ).
% set_to_map_simp
thf(fact_5043_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_5044_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S )
= ( ^ [X3: A] : ( none @ B ) ) )
= ( S
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_5045_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( set_to_map @ A @ B @ S ) )
= ( S
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(2)
thf(fact_5046_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_5047_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_5048_lexn_Osimps_I1_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R3 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_5049_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_5050_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_5051_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 ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_5052_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_5053_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X3: A,Y3: B,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_5054_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 @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_5055_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y3: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X3: A,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_5056_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_5057_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_5058_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_5059_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( member @ ( 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
thf(fact_5060_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( 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
thf(fact_5061_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( 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
thf(fact_5062_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( 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
thf(fact_5063_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X2: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Y2: B,Ys4: list @ B] :
( ( Ys
= ( cons @ B @ Y2 @ Ys4 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X2 @ Y2 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_5064_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H3: B > C > A,F2: D > B,Xs: list @ D,G2: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H3 ) @ ( zip @ B @ C @ ( map @ D @ B @ F2 @ Xs ) @ ( map @ D @ C @ G2 @ Xs ) ) )
= ( map @ D @ A
@ ^ [X3: D] : ( H3 @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_5065_set__zip__cart,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L4: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L4 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu: A] : ( set2 @ B @ L4 ) ) ) ) ).
% set_zip_cart
thf(fact_5066_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs2: list @ A,Ys2: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) )
@ ( zip @ B @ A @ Ys2 @ Xs2 ) ) ) ) ).
% zip_commute
thf(fact_5067_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X2: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_5068_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y2: B] :
~ ( member @ ( 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
thf(fact_5069_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M2: A > ( option @ B ),Ks2: list @ A,Vs3: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N2: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K4: A,V2: B] : ( fun_upd @ A @ ( option @ B ) @ N2 @ K4 @ ( some @ B @ V2 ) ) )
@ M2
@ ( zip @ A @ B @ Ks2 @ Vs3 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_5070_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 ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_5071_nths__shift__lemma,axiom,
! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_5072_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: ( product_prod @ B @ C ) > A,Xs: list @ B,G2: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G2 @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X3: B,Y3: D] : ( F2 @ ( product_Pair @ B @ C @ X3 @ ( G2 @ Y3 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_5073_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: ( product_prod @ B @ C ) > A,G2: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ ( map @ D @ B @ G2 @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X3: D,Y3: C] : ( F2 @ ( product_Pair @ B @ C @ ( G2 @ X3 ) @ Y3 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_5074_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list @ A,Xs: list @ B,F2: C > A > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B4: C] :
( product_case_prod @ B @ A @ C
@ ^ [X3: B] : ( F2 @ B4 ) )
@ B2
@ ( zip @ B @ A @ Xs @ Ys ) )
= ( foldl @ C @ A @ F2 @ B2 @ Ys ) ) ) ).
% foldl_snd_zip
thf(fact_5075_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R3 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_5076_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_5077_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list @ B] :
( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L )
= ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) @ L ) ) ).
% map_zip2
thf(fact_5078_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs2: list @ A,A6: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A6 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_5079_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 )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_5080_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G2: D > B,Xs: list @ C,Ys: list @ D] :
( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F2 @ X3 ) @ ( G2 @ Y3 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ ( map @ D @ B @ G2 @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_5081_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F2: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F2 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X3: A,Y3: C] : ( product_Pair @ A @ B @ X3 @ ( F2 @ Y3 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_5082_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F2 @ X3 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_5083_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z5: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z5 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_5084_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y3: B,Ys2: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_5085_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remdups_adj @ A @ Xs )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y3: A] :
( Y3
= ( last @ A @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_5086_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_5087_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_5088_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_5089_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list @ A,Xs: list @ B,F2: A > C > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X3: B] : F2 )
@ ( zip @ B @ A @ Xs @ Ys )
@ B2 )
= ( foldr @ A @ C @ F2 @ Ys @ B2 ) ) ) ).
% foldr_snd_zip
thf(fact_5090_lexn_Osimps_I2_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R3 @ ( suc @ N ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R3 @ ( lexn @ A @ R3 @ N ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= ( suc @ N ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_5091_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [I3: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_5092_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,Z5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R3 )
@ X3 ) ) ) ) ).
% listrel_iff_zip
thf(fact_5093_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [Y3: B] : Y3 )
= ( ^ [Z5: product_prod @ A @ B] : Z5 ) ) ).
% map_prod_ident
thf(fact_5094_ball__empty,axiom,
! [A: $tType,P: A > $o,X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_5095_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G2: D > B,A3: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 @ ( product_Pair @ C @ D @ A3 @ B2 ) )
= ( product_Pair @ A @ B @ ( F2 @ A3 ) @ ( G2 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_5096_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P: A > $o,Q2: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
? [Y3: A] :
( ( P @ Y3 )
& ( Q2 @ X3 @ Y3 ) ) ) )
= ( ! [Y3: A] :
( ( P @ Y3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] : ( Q2 @ X3 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_5097_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: B,B3: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [L2: B] :
( ( Uu
= ( F2 @ L2 ) )
& ( ( L2 = A3 )
| ( member @ B @ L2 @ B3 ) ) ) )
= ( insert2 @ A @ ( F2 @ A3 )
@ ( collect @ A
@ ^ [Uu: A] :
? [L2: B] :
( ( Uu
= ( F2 @ L2 ) )
& ( member @ B @ L2 @ B3 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_5098_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),F2: A > C,G2: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ A3 ) @ ( G2 @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F2 @ G2 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_5099_INF__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A6: set @ A,F: A > $o] : ( complete_Inf_Inf @ $o @ ( image2 @ A @ $o @ F @ A6 ) ) )
= ( ball @ A ) ) ).
% INF_bool_eq
thf(fact_5100_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F2: B > A,P: B > B > $o] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F2 ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A5: B,B4: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F2 @ A5 ) @ ( F2 @ B4 ) ) )
& ( P @ A5 @ B4 ) ) ) ) ).
% pairself_image_eq
thf(fact_5101_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q2: B > $o,F2: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q2 ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu: C] :
? [X3: A,Y3: B] :
( ( Uu
= ( F2 @ X3 @ Y3 ) )
& ( P @ X3 )
& ( Q2 @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_5102_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu: B] :
? [X3: A] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_5103_Ball__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K4: A,S2: $o] :
( S2
& ( P @ K4 ) )
@ $true
@ A4 ) ) ) ).
% Ball_fold
thf(fact_5104_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X3: set @ A] :
( ( X3
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,A5: A] :
( ( X3
= ( insert2 @ A @ A5 @ A6 ) )
& ( P6 @ A6 ) ) ) ) ).
% finite.mono
thf(fact_5105_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B > C,S: set @ C] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [P6: product_prod @ A @ B] :
( ( Uu = P6 )
& ( member @ C @ ( F2 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) @ S ) ) ) )
= ( collect @ A
@ ^ [A5: A] :
? [B4: B] : ( member @ C @ ( F2 @ A5 @ B4 ) @ S ) ) ) ).
% fs_contract
thf(fact_5106_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X3: B] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image2 @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5107_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X3: B] :
( ( Uu
= ( F2 @ X3 ) )
& ( member @ B @ X3 @ A4 ) ) )
= ( image2 @ B @ A @ F2 @ A4 ) ) ).
% Setcompr_eq_image
thf(fact_5108_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H3: B > C > A,F2: D > B,G2: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H3 @ ( product_map_prod @ D @ B @ E @ C @ F2 @ G2 @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L2: D,R4: E] : ( H3 @ ( F2 @ L2 ) @ ( G2 @ R4 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_5109_Union__SetCompr__eq,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A ),P: B > $o] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [X3: B] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) ) )
= ( collect @ A
@ ^ [A5: A] :
? [X3: B] :
( ( P @ X3 )
& ( member @ A @ A5 @ ( F2 @ X3 ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_5110_rtranclp_Omono,axiom,
! [A: $tType,R3: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P6: A > A > $o,X12: A,X23: A] :
( ? [A5: A] :
( ( X12 = A5 )
& ( X23 = A5 ) )
| ? [A5: A,B4: A,C5: A] :
( ( X12 = A5 )
& ( X23 = C5 )
& ( P6 @ A5 @ B4 )
& ( R3 @ B4 @ C5 ) ) ) ) ).
% rtranclp.mono
thf(fact_5111_tranclp_Omono,axiom,
! [A: $tType,R3: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P6: A > A > $o,X12: A,X23: A] :
( ? [A5: A,B4: A] :
( ( X12 = A5 )
& ( X23 = B4 )
& ( R3 @ A5 @ B4 ) )
| ? [A5: A,B4: A,C5: A] :
( ( X12 = A5 )
& ( X23 = C5 )
& ( P6 @ A5 @ B4 )
& ( R3 @ B4 @ C5 ) ) ) ) ).
% tranclp.mono
thf(fact_5112_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3
@ T4 )
= T4 ) ).
% prod.map_ident
thf(fact_5113_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X23: list @ A] :
( ? [Y3: A,Ys2: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( P6 @ Xs2 @ Ys2 ) ) ) ) ) ).
% lexordp.mono
thf(fact_5114_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A6: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z5: list @ A] :
? [X3: A,Xs2: list @ A] :
( ( Z5
= ( cons @ A @ X3 @ Xs2 ) )
& ( member @ A @ X3 @ A6 )
& ( member @ ( list @ A ) @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_5115_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X23: list @ A] :
( ? [Y3: A,Ys2: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ( Less @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ~ ( Less @ X3 @ Y3 )
& ~ ( Less @ Y3 @ X3 )
& ( P6 @ Xs2 @ Ys2 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_5116_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_5117_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X3: B] :
( U2
= ( F2 @ X3 ) ) )
= ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5118_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A3: A,A4: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ A4 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [B4: A] :
( ( Uu
= ( inf_inf @ A @ A3 @ B4 ) )
& ( member @ A @ B4 @ A4 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_5119_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F2: C > A,G2: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 ) @ R ) )
=> ~ ! [X2: C,Y2: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G2 @ Y2 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y2 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_5120_Field__not__elem,axiom,
! [A: $tType,V: A,R: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ V @ ( field2 @ A @ R ) )
=> ! [X5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X5 @ R )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z5: A] :
( ( Y3 != V )
& ( Z5 != V ) )
@ X5 ) ) ) ).
% Field_not_elem
thf(fact_5121_INTER__eq,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) )
= ( collect @ A
@ ^ [Y3: A] :
! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( member @ A @ Y3 @ ( B3 @ X3 ) ) ) ) ) ).
% INTER_eq
thf(fact_5122_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A4: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X3: A] :
! [Y3: B] :
( ( member @ B @ Y3 @ A4 )
=> ( P @ X3 @ Y3 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] :
( collect @ A
@ ^ [X3: A] : ( P @ X3 @ Y3 ) )
@ A4 ) ) ) ).
% Collect_ball_eq
thf(fact_5123_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P6: product_prod @ A @ A] :
? [X3: A] :
( P6
= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_def
thf(fact_5124_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F: A > C,G: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X3: A,Y3: B] : ( product_Pair @ C @ D @ ( F @ X3 ) @ ( G @ Y3 ) ) ) ) ) ).
% map_prod_def
thf(fact_5125_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ B
@ ^ [B4: B] :
? [A5: A] :
( ( M2 @ A5 )
= ( some @ B @ B4 ) ) ) ) ) ).
% ran_def
thf(fact_5126_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A6: set @ A,F: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A5: A] :
( ( Uu
= ( product_Pair @ A @ B @ A5 @ ( F @ A5 ) ) )
& ( member @ A @ A5 @ A6 ) ) ) ) ) ).
% Gr_def
thf(fact_5127_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A4: B > ( set @ C ),F2: C > $o,G2: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X3: B] :
! [Y3: C] :
( ( member @ C @ Y3 @ ( A4 @ X3 ) )
=> ( F2 @ Y3 ) )
@ G2 )
= ( ^ [X3: A] :
! [Y3: C] :
( ( member @ C @ Y3 @ ( comp @ B @ ( set @ C ) @ A @ A4 @ G2 @ X3 ) )
=> ( F2 @ Y3 ) ) ) ) ).
% Ball_comp_iff
thf(fact_5128_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
? [Us3: list @ A,Z5: A,Z7: A,Vs3: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ ( cons @ A @ Z5 @ Vs3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ Z7 ) @ R4 )
& ( Ys2
= ( append @ A @ Us3 @ ( cons @ A @ Z7 @ Vs3 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_5129_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X3: list @ A,Y3: list @ A] :
? [A5: A,V2: list @ A] :
( ( Y3
= ( append @ A @ X3 @ ( cons @ A @ A5 @ V2 ) ) )
| ? [U2: list @ A,B4: A,C5: A,W3: list @ A,Z5: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R4 )
& ( X3
= ( append @ A @ U2 @ ( cons @ A @ B4 @ W3 ) ) )
& ( Y3
= ( append @ A @ U2 @ ( cons @ A @ C5 @ Z5 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_5130_case__prod__o__map__prod,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F2: D > E > C,G1: A > D,G22: B > E] :
( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F2 ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
= ( product_case_prod @ A @ B @ C
@ ^ [L2: A,R4: B] : ( F2 @ ( G1 @ L2 ) @ ( G22 @ R4 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_5131_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P2: A > $o,X3: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P2 @ Y3 ) ) ) ) ).
% ID.pred_set
thf(fact_5132_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B15: A,B24: A,B33: A,F2: A > B] :
( ( ord_less_eq @ A @ B15 @ B24 )
=> ( ( ord_less_eq @ A @ B24 @ B33 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [I3: A] :
( ( Uu
= ( F2 @ I3 ) )
& ( ord_less_eq @ A @ B15 @ I3 )
& ( ord_less @ A @ I3 @ B24 ) ) )
@ ( collect @ B
@ ^ [Uu: B] :
? [I3: A] :
( ( Uu
= ( F2 @ I3 ) )
& ( ord_less_eq @ A @ B24 @ I3 )
& ( ord_less @ A @ I3 @ B33 ) ) ) )
= ( collect @ B
@ ^ [Uu: B] :
? [I3: A] :
( ( Uu
= ( F2 @ I3 ) )
& ( ord_less_eq @ A @ B15 @ I3 )
& ( ord_less @ A @ I3 @ B33 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_5133_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ B4 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5134_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ B4 @ X3 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5135_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_5136_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X3: A] :
? [X4: B] : ( P @ X3 @ X4 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] :
( collect @ A
@ ^ [X3: A] : ( P @ X3 @ Y3 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_5137_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),N2: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N2 )
& ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_5138_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R4: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Z5: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R4 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z5 ) @ S2 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_5139_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_5140_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_5141_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A5: A,B4: B] :
( ( Uu
= ( product_Pair @ A @ B @ A5 @ B4 ) )
& ( ( M2 @ A5 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_5142_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: B > A,A4: set @ B,A17: set @ A,G2: D > C,B3: set @ D,B16: set @ C] :
( ( ( image2 @ B @ A @ F2 @ A4 )
= A17 )
=> ( ( ( image2 @ D @ C @ G2 @ B3 )
= B16 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F2 @ G2 )
@ ( product_Sigma @ B @ D @ A4
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ C @ A17
@ ^ [Uu: A] : B16 ) ) ) ) ).
% map_prod_surj_on
thf(fact_5143_set__map__filter,axiom,
! [B: $tType,A: $tType,G2: B > ( option @ A ),Xs: list @ B] :
( ( set2 @ A @ ( map_filter @ B @ A @ G2 @ Xs ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
& ( ( G2 @ X3 )
= ( some @ A @ Y3 ) ) ) ) ) ).
% set_map_filter
thf(fact_5144_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,G2: C > D,B3: set @ C] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( inj_on @ C @ D @ G2 @ B3 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F2 @ G2 )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : B3 ) ) ) ) ).
% map_prod_inj_on
thf(fact_5145_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs2: list @ A] :
( collect @ A
@ ^ [Uu: A] :
? [I3: nat] :
( ( Uu
= ( nth @ A @ Xs2 @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5146_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A,B4: A] :
( ( Uu
= ( inf_inf @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B4 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_5147_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A] :
( ( Uu
= ( inf_inf @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_5148_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A,B4: A] :
( ( Uu
= ( sup_sup @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B4 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_5149_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A] :
( ( Uu
= ( sup_sup @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_5150_wf__map__prod__image,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( wf @ A @ R3 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( wf @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F2 @ F2 ) @ R3 ) ) ) ) ).
% wf_map_prod_image
thf(fact_5151_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z4: list @ A] : Y5 = Z4 )
= ( ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_5152_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: A > B,G2: C > D] :
( ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G2 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F2 @ G2 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_5153_prod_Oinj__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F22: B > D] :
( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ B @ D @ F22 @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% prod.inj_map
thf(fact_5154_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y3: list @ A,Z5: list @ A] :
( ( size_size @ ( list @ A ) @ Y3 )
= ( size_size @ ( list @ A ) @ Z5 ) )
@ X2 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_5155_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y3: list @ A,Z5: list @ A] :
( ( size_size @ ( list @ A ) @ Y3 )
= ( size_size @ ( list @ A ) @ Z5 ) )
@ X2 ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_5156_set__nths,axiom,
! [A: $tType,Xs: list @ A,I4: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I4 ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I3: nat] :
( ( Uu
= ( nth @ A @ Xs @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I4 ) ) ) ) ).
% set_nths
thf(fact_5157_funpow__inj__finite,axiom,
! [A: $tType,P4: A > A,X: A] :
( ( inj_on @ A @ A @ P4 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] :
? [N2: nat] :
( Y3
= ( compow @ ( A > A ) @ N2 @ P4 @ X ) ) ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( compow @ ( A > A ) @ N3 @ P4 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5158_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N @ L ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I3: nat] :
( ( Uu
= ( nth @ A @ L @ I3 ) )
& ( ord_less_eq @ nat @ N @ I3 )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_5159_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,G2: C > B,A4: set @ A,B3: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A5: A,B4: C] : ( times_times @ B @ ( F2 @ A5 ) @ ( G2 @ B4 ) ) )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : B3 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [A5: A,B4: C] :
( ( Uu
= ( times_times @ B @ ( F2 @ A5 ) @ ( G2 @ B4 ) ) )
& ( member @ A @ A5 @ A4 )
& ( member @ C @ B4 @ B3 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_5160_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R4 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( B4 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X3 ) @ R4 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_5161_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R4 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X3 ) @ R4 ) ) ) ) ) ) ).
% Under_def
thf(fact_5162_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys3 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys3 ) )
=> ( X @ X2 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_5163_Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F @ A4 ) )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( member @ A @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% Inf_Sup
thf(fact_5164_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A4 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F @ A4 ) )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( member @ A @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_5165_Inter__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ A
@ ^ [X3: A] :
! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ A6 )
=> ( member @ A @ X3 @ Y3 ) ) ) ) ) ).
% Inter_eq
thf(fact_5166_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q2: A > B > C,F2: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q2 )
=> ( order_mono @ A @ ( D > C )
@ ^ [I3: A,X3: D] : ( Q2 @ I3 @ ( F2 @ X3 ) ) ) ) ) ).
% mono_compose
thf(fact_5167_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F @ A4 ) )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( member @ A @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5168_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F @ A4 ) )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( member @ A @ ( F @ X3 ) @ X3 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5169_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F @ A4 ) )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( member @ A @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5170_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G2: B > A,A4: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X3: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ X3 ) )
@ A4 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X3: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ X3 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F @ A4 ) )
& ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ A4 )
=> ( member @ B @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_5171_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G2: B > A,A4: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B5: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B5 ) )
@ A4 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B5: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B5 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F @ A4 ) )
& ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ A4 )
=> ( member @ B @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_5172_option__Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ ( option @ A ) )] :
( ord_less_eq @ ( option @ A ) @ ( complete_Inf_Inf @ ( option @ A ) @ ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ ( complete_Sup_Sup @ ( option @ A ) ) @ A4 ) )
@ ( complete_Sup_Sup @ ( option @ A )
@ ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ ( complete_Inf_Inf @ ( option @ A ) )
@ ( collect @ ( set @ ( option @ A ) )
@ ^ [Uu: set @ ( option @ A )] :
? [F: ( set @ ( option @ A ) ) > ( option @ A )] :
( ( Uu
= ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ F @ A4 ) )
& ! [X3: set @ ( option @ A )] :
( ( member @ ( set @ ( option @ A ) ) @ X3 @ A4 )
=> ( member @ ( option @ A ) @ ( F @ X3 ) @ X3 ) ) ) ) ) ) ) ) ).
% option_Inf_Sup
thf(fact_5173_Pow__Compl,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [B5: set @ A] :
( ( Uu
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
& ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B5 ) ) ) ) ) ).
% Pow_Compl
thf(fact_5174_Union__maximal__sets,axiom,
! [A: $tType,F8: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F8 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T7: set @ A] :
( ( member @ ( set @ A ) @ T7 @ F8 )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ F8 )
=> ~ ( ord_less @ ( set @ A ) @ T7 @ X3 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F8 ) ) ) ).
% Union_maximal_sets
thf(fact_5175_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S8: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F7: filter @ A] :
! [X3: filter @ A] :
( ( member @ ( filter @ A ) @ X3 @ S8 )
=> ( ord_less_eq @ ( filter @ A ) @ F7 @ X3 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_5176_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu: A] :
? [N2: int] :
( ( Uu
= ( ring_1_of_int @ A @ N2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_5177_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X4: set @ int] :
( the @ int
@ ^ [X3: int] :
( ( member @ int @ X3 @ X4 )
& ! [Y3: int] :
( ( member @ int @ Y3 @ X4 )
=> ( ord_less_eq @ int @ Y3 @ X3 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_5178_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys3 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys3 ) )
=> ( X @ X2 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_5179_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys3 ) )
=> ( ( Y
= ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ( X @ X2 @ Y3 ) )
& ( sorted_wrt @ A @ X @ Ys3 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_5180_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R4 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B4 ) @ R4 ) ) ) ) ) ) ).
% Above_def
thf(fact_5181_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V )
@ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [V6: B] :
( Uu
= ( product_Pair @ A @ B @ K @ V6 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_5182_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X3: A,Y3: B] :
( ( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X3: A,Y3: B] :
( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_5183_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_5184_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M ) )
= ( M
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_5185_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_5186_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( M2 @ K4 )
= ( some @ B @ V2 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_5187_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R2: A > B > $o,P2: B > $o,X3: A] :
? [Y3: B] :
( ( R2 @ X3 @ Y3 )
& ( P2 @ Y3 ) ) ) ) ).
% rel_pred_comp_def
thf(fact_5188_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P6: A > $o,X3: A] :
( ? [Y3: A] :
( ( X3
= ( F2 @ Y3 ) )
& ( P6 @ Y3 ) )
| ? [M9: set @ A] :
( ( X3
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ M9 )
=> ( P6 @ Y3 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_5189_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [I3: nat] :
( ( Uu
= ( F @ ( nth @ B @ L2 @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_5190_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F2 @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_5191_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A3: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F2 @ ( cons @ B @ A3 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ A3 ) @ ( list_collect_set @ B @ A @ F2 @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_5192_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),L: list @ B,L4: list @ B] :
( ( list_collect_set @ B @ A @ F2 @ ( append @ B @ L @ L4 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F2 @ L ) @ ( list_collect_set @ B @ A @ F2 @ L4 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_5193_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > ( set @ A ),X: C > B] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_5194_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,A3: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ ( X @ A3 ) ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_5195_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,L: list @ C,L4: list @ C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( append @ C @ L @ L4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L4 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_5196_chain__compr,axiom,
! [A: $tType,Ord: A > A > $o,A4: set @ A,P: A > $o] :
( ( comple1602240252501008431_chain @ A @ Ord @ A4 )
=> ( comple1602240252501008431_chain @ A @ Ord
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ).
% chain_compr
thf(fact_5197_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_5198_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_5199_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [A5: B] :
( ( Uu
= ( F @ A5 ) )
& ( member @ B @ A5 @ ( set2 @ B @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_5200_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% in_chain_finite
thf(fact_5201_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A6: set @ C,F: C > A,G: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A5: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F @ A5 ) @ ( G @ A5 ) ) )
& ( member @ C @ A5 @ A6 ) ) ) ) ) ).
% image2_def
thf(fact_5202_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X3: set @ A] :
( ( X3
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,A5: A] :
( ( X3
= ( insert2 @ A @ A5 @ A6 ) )
& ( P6 @ A6 ) ) ) ) ) ).
% finite_def
thf(fact_5203_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A > A] :
( ! [X2: A,Y2: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 @ W ) @ ( F2 @ Y2 @ Z3 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X3: A] : ( complete_lattice_lfp @ A @ ( F2 @ X3 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X3: A] : ( F2 @ X3 @ X3 ) ) ) ) ) ).
% lfp_lfp
thf(fact_5204_lfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T4: A] :
( ( complete_lattice_lfp @ A
@ ^ [X3: A] : T4 )
= T4 ) ) ).
% lfp_const
thf(fact_5205_lfp__rolling,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [G2: A > B,F2: B > A] :
( ( order_mono @ A @ B @ G2 )
=> ( ( order_mono @ B @ A @ F2 )
=> ( ( G2
@ ( complete_lattice_lfp @ A
@ ^ [X3: A] : ( F2 @ ( G2 @ X3 ) ) ) )
= ( complete_lattice_lfp @ B
@ ^ [X3: B] : ( G2 @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_5206_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_5207_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F2: B > A,X: B,C2: C,G2: B > C,A4: set @ B] :
( ( B2
= ( F2 @ X ) )
=> ( ( C2
= ( G2 @ X ) )
=> ( ( member @ B @ X @ A4 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A4 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_5208_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F2: A > A,P: A] :
( ( A4
= ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ ( inf_inf @ A @ A4 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A4 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_5209_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,P: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F2 ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_5210_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_5211_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P6: A > $o,X3: A] :
( ? [Y3: A] :
( ( X3
= ( F @ Y3 ) )
& ( P6 @ Y3 ) )
| ? [M9: set @ A] :
( ( X3
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ M9 )
=> ( P6 @ Y3 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_5212_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B4: A,A6: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4
@ ( the @ A
@ ^ [X3: A] : ( member @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_5213_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_Rats @ A )
= ( image2 @ nat @ A @ ( comp @ rat @ A @ nat @ ( field_char_0_of_rat @ A ) @ nat_to_rat_surj ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Rats_eq_range_of_rat_o_nat_to_rat_surj
thf(fact_5214_Rats__minus__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( field_char_0_Rats @ A ) )
= ( member @ A @ A3 @ ( field_char_0_Rats @ A ) ) ) ) ).
% Rats_minus_iff
thf(fact_5215_Rats__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_1
thf(fact_5216_Rats__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_mult
thf(fact_5217_def__lfp__induct__set,axiom,
! [A: $tType,A4: set @ A,F2: ( set @ A ) > ( set @ A ),A3: A,P: A > $o] :
( ( A4
= ( complete_lattice_lfp @ ( set @ A ) @ F2 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( F2 @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_5218_lfp__induct__set,axiom,
! [A: $tType,A3: A,F2: ( set @ A ) > ( set @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( complete_lattice_lfp @ ( set @ A ) @ F2 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( F2 @ ( inf_inf @ ( set @ A ) @ ( complete_lattice_lfp @ ( set @ A ) @ F2 ) @ ( collect @ A @ P ) ) ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ) ).
% lfp_induct_set
thf(fact_5219_Rats__def,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_Rats @ A )
= ( image2 @ rat @ A @ ( field_char_0_of_rat @ A ) @ ( top_top @ ( set @ rat ) ) ) ) ) ).
% Rats_def
thf(fact_5220_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F2: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F2 ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F2 )
=> ( ! [A8: A,B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( F2 @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A8 @ B7 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_5221_Rats__eq__range__nat__to__rat__surj,axiom,
( ( field_char_0_Rats @ rat )
= ( image2 @ nat @ rat @ nat_to_rat_surj @ ( top_top @ ( set @ nat ) ) ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_5222_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X23: list @ A] :
( ? [Y3: A,Ys2: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( P6 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_5223_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( dom @ A @ B @ ( F2 @ A4 ) ) )
=> ( ( A4
= ( restrict_map @ A @ B @ ( F2 @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S3 ) ) )
= ( ( map_le @ A @ B @ A4 @ ( F2 @ A4 ) )
& ( S3
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F2 @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_5224_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( dom @ A @ B @ ( F2 @ A4 ) ) )
& ( A4
= ( restrict_map @ A @ B @ ( F2 @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S3 ) ) ) )
= ( ( map_le @ A @ B @ A4 @ ( F2 @ A4 ) )
& ( S3
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F2 @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_5225_map__le__empty,axiom,
! [B: $tType,A: $tType,G2: A > ( option @ B )] :
( map_le @ A @ B
@ ^ [X3: A] : ( none @ B )
@ G2 ) ).
% map_le_empty
thf(fact_5226_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs2: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_5227_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X23: list @ A] :
( ? [Y3: A,Ys2: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( X12
= ( cons @ A @ X3 @ Xs2 ) )
& ( X23
= ( cons @ A @ Y3 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( P6 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_5228_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X4: set @ A,Y9: set @ A] :
( ( Uu
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y9 ) )
& ( X4
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ Y9 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_5229_lists__empty,axiom,
! [A: $tType] :
( ( lists @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% lists_empty
thf(fact_5230_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_5231_lists__Int__eq,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( lists @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ A4 ) @ ( lists @ A @ B3 ) ) ) ).
% lists_Int_eq
thf(fact_5232_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A4: set @ A,Q2: B > A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( Q2 @ X3 @ Y3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y3: B] : ( Q2 @ Y3 @ X3 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_5233_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X4: A] : ( P @ X4 ) ) ) ).
% bex_UNIV
thf(fact_5234_lists__UNIV,axiom,
! [A: $tType] :
( ( lists @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% lists_UNIV
thf(fact_5235_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o,A4: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) @ A4 )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ X3 @ Y3 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_5236_SUP__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A6: set @ A,F: A > $o] : ( complete_Sup_Sup @ $o @ ( image2 @ A @ $o @ F @ A6 ) ) )
= ( bex @ A ) ) ).
% SUP_bool_eq
thf(fact_5237_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F: A > B,A6: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( Y3
= ( F @ X3 ) ) ) ) ) ) ).
% image_def
thf(fact_5238_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A6: set @ A,P2: A > $o] :
? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def_raw
thf(fact_5239_Union__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A6: set @ ( set @ A )] :
( collect @ A
@ ^ [X3: A] :
? [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ A6 )
& ( member @ A @ X3 @ Y3 ) ) ) ) ) ).
% Union_eq
thf(fact_5240_lists__IntI,axiom,
! [A: $tType,L: list @ A,A4: set @ A,B3: set @ A] :
( ( member @ ( list @ A ) @ L @ ( lists @ A @ A4 ) )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ B3 ) )
=> ( member @ ( list @ A ) @ L @ ( lists @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% lists_IntI
thf(fact_5241_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),S2: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).
% Image_def
thf(fact_5242_UNION__eq,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A ),A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ A4 ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( member @ A @ Y3 @ ( B3 @ X3 ) ) ) ) ) ).
% UNION_eq
thf(fact_5243_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A4: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X3: A] :
? [Y3: B] :
( ( member @ B @ Y3 @ A4 )
& ( P @ X3 @ Y3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] :
( collect @ A
@ ^ [X3: A] : ( P @ X3 @ Y3 ) )
@ A4 ) ) ) ).
% Collect_bex_eq
thf(fact_5244_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( vimage @ A @ B @ F2 @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( F2 @ X3 )
= ( F2 @ Y3 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_5245_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A6: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A6 ) ) ) ) ).
% lists_eq_set
thf(fact_5246_Bex__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K4: A,S2: $o] :
( S2
| ( P @ K4 ) )
@ $false
@ A4 ) ) ) ).
% Bex_fold
thf(fact_5247_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A4: set @ nat,B3: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A4 ) @ B3 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I3: nat] :
( ( member @ nat @ I3 @ A4 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I8: nat] :
( ( member @ nat @ I8 @ A4 )
& ( ord_less @ nat @ I8 @ I3 ) ) ) )
@ B3 ) ) ) ) ) ).
% nths_nths
thf(fact_5248_max__extp_Omax__extI,axiom,
! [A: $tType,X7: set @ A,Y4: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y4 )
=> ( ( Y4
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y4 )
& ( R @ X2 @ Xa2 ) ) )
=> ( max_extp @ A @ R @ X7 @ Y4 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_5249_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R2: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A12 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A23 )
& ( R2 @ X3 @ Y3 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_5250_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R @ X5 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_5251_listrel__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R3 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A4 )
@ ^ [Uu: list @ A] : ( lists @ A @ A4 ) ) ) ) ).
% listrel_subset
thf(fact_5252_lists__of__len__fin2,axiom,
! [A: $tType,P: set @ A,N: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L2: list @ A] :
( N
= ( size_size @ ( list @ A ) @ L2 ) ) ) ) ) ) ).
% lists_of_len_fin2
thf(fact_5253_lists__of__len__fin1,axiom,
! [A: $tType,P: set @ A,N: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L2: list @ A] :
( ( size_size @ ( list @ A ) @ L2 )
= N ) ) ) ) ) ).
% lists_of_len_fin1
thf(fact_5254_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F: A > ( option @ B ),A6: set @ A] :
( collect @ B
@ ^ [B4: B] :
? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ( F @ X3 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% map_project_def
thf(fact_5255_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X4: set @ A,Y9: set @ A] :
( ( finite_finite2 @ A @ X4 )
& ( finite_finite2 @ A @ Y9 )
& ( Y9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ Y9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_5256_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q2: A > A] :
( ( order_mono @ A @ A @ Q2 )
=> ( order_antimono @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q2 @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_5257_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X4: A] : ( P @ X4 ) ) ) ).
% ball_UNIV
thf(fact_5258_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A6: A > C > $o,B5: B > D > $o,F: A > B,G: C > D] :
! [X3: A,Y3: C] :
( ( A6 @ X3 @ Y3 )
=> ( B5 @ ( F @ X3 ) @ ( G @ Y3 ) ) ) ) ) ).
% rel_fun_def
thf(fact_5259_rel__fun__eq__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R )
= ( ^ [F: A > B,G: A > C] :
! [X3: A] : ( R @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% rel_fun_eq_rel
thf(fact_5260_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P2: A > $o] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( P2 @ X3 ) ) ) ) ).
% Ball_def_raw
thf(fact_5261_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_5262_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_5263_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_5264_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F: A > B] :
! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) ) ) ).
% antimono_def
thf(fact_5265_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B,D3: B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F: A > B] :
! [X3: A] :
( ( ( member @ A @ X3 @ A4 )
=> ( member @ B @ ( F @ X3 ) @ B3 ) )
& ( ~ ( member @ A @ X3 @ A4 )
=> ( ( F @ X3 )
= D3 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_5266_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X3: A] :
! [X4: B] : ( P @ X3 @ X4 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] :
( collect @ A
@ ^ [X3: A] : ( P @ X3 @ Y3 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_5267_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P2 @ X3 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_5268_Func__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_Func @ A @ B )
= ( ^ [A6: set @ A,B5: set @ B] :
( collect @ ( A > B )
@ ^ [F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F @ X3 ) @ B5 ) )
& ! [A5: A] :
( ~ ( member @ A @ A5 @ A6 )
=> ( ( F @ A5 )
= ( undefined @ B ) ) ) ) ) ) ) ).
% Func_def
thf(fact_5269_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_min @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
= ( F2 @ ( ord_max @ A @ X @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_5270_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_max @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
= ( F2 @ ( ord_min @ A @ X @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_5271_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P2 @ X3 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5272_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P2: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P2 @ X3 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Less_eq2 @ X3 @ Y3 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5273_transfer__bforall__def,axiom,
! [A: $tType] :
( ( transfer_bforall @ A )
= ( ^ [P2: A > $o,Q: A > $o] :
! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% transfer_bforall_def
thf(fact_5274_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5275_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_5276_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X2 ) )
=> ( Q2 @ X2 ) ) )
=> ( Q2 @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5277_listrel1__subset__listrel,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R5 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R3 ) @ ( listrel @ A @ A @ R5 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_5278_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X212: A,X223: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X212 @ X223 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X212 ) @ ( size_list @ A @ X @ X223 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_5279_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ Xs @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_5280_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F: A > B > A,S2: A,Xs2: list @ B] :
( fold @ B @ A
@ ^ [X3: B,T3: A] : ( F @ T3 @ X3 )
@ Xs2
@ S2 ) ) ) ).
% foldl_conv_fold
thf(fact_5281_refl__onD2,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_5282_refl__onD1,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_5283_refl__onD,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A4 @ R3 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_5284_refl__on__domain,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A4 )
& ( member @ A @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_5285_refl__rtrancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) ) ).
% refl_rtrancl
thf(fact_5286_refl__on__empty,axiom,
! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% refl_on_empty
thf(fact_5287_refl__on__Int,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),B3: set @ A,S3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R3 )
=> ( ( refl_on @ A @ B3 @ S3 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) ) ) ) ).
% refl_on_Int
thf(fact_5288_refl__Id,axiom,
! [A: $tType] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( id2 @ A ) ) ).
% refl_Id
thf(fact_5289_refl__on__Un,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),B3: set @ A,S3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R3 )
=> ( ( refl_on @ A @ B3 @ S3 )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) ) ) ) ).
% refl_on_Un
thf(fact_5290_fold__filter,axiom,
! [A: $tType,B: $tType,F2: B > A > A,P: B > $o,Xs: list @ B] :
( ( fold @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( fold @ B @ A
@ ^ [X3: B] : ( if @ ( A > A ) @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( id @ A ) )
@ Xs ) ) ).
% fold_filter
thf(fact_5291_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
= ( fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ Xs @ A4 ) ) ).
% union_set_fold
thf(fact_5292_refl__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ).
% refl_reflcl
thf(fact_5293_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 )
@ Xs
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_5294_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R4 ) ) ) ) ) ).
% refl_on_def
thf(fact_5295_refl__onI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 ) )
=> ( refl_on @ A @ A4 @ R3 ) ) ) ).
% refl_onI
thf(fact_5296_Refl__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( refl_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Refl_Restr
thf(fact_5297_Sup__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Sup_Sup @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% Sup_set_fold
thf(fact_5298_Inf__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Inf_Inf @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ ( top_top @ A ) ) ) ) ).
% Inf_set_fold
thf(fact_5299_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z5: A] :
( ( member @ A @ Y3 @ A6 )
& ( member @ A @ Z5 @ A6 ) )
@ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R4 ) ) ) ) ) ).
% refl_on_def'
thf(fact_5300_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ X ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_5301_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_5302_refl__on__reflcl__Image,axiom,
! [A: $tType,B3: set @ A,A4: set @ ( product_prod @ A @ A ),C3: set @ A] :
( ( refl_on @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B3 )
=> ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) @ C3 )
= ( image @ A @ A @ A4 @ C3 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_5303_Refl__Field__Restr2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
= A4 ) ) ) ).
% Refl_Field_Restr2
thf(fact_5304_Refl__Field__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( field2 @ A @ R3 ) @ A4 ) ) ) ).
% Refl_Field_Restr
thf(fact_5305_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_5306_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_5307_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_5308_rotate1__length01,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_5309_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R2: A > B > $o,A5: list @ A,B4: list @ B] :
? [Z5: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A5 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B4 ) ) ) ) ).
% list.in_rel
thf(fact_5310_antisym__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( antisym @ A @ R3 ) ) ).
% antisym_reflcl
thf(fact_5311_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 )
=> ( X3 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_5312_antisymI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R3 )
=> ( X2 = Y2 ) ) )
=> ( antisym @ A @ R3 ) ) ).
% antisymI
thf(fact_5313_antisymD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( A3 = B2 ) ) ) ) ).
% antisymD
thf(fact_5314_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As: list @ A,F2: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As @ ( map @ C @ B @ F2 @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X3: A,Y3: C] : ( P @ X3 @ ( F2 @ Y3 ) )
@ As
@ Bs ) ) ).
% list_all2_map2
thf(fact_5315_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F2: C > A,As: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F2 @ As ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X3: C] : ( P @ ( F2 @ X3 ) )
@ As
@ Bs ) ) ).
% list_all2_map1
thf(fact_5316_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I @ X ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% list.rel_map(1)
thf(fact_5317_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: list @ A,G2: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X @ ( map @ B @ C @ G2 @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G2 @ Y3 ) )
@ X
@ Y ) ) ).
% list.rel_map(2)
thf(fact_5318_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [List: list @ A] :
( List
!= ( nil @ A ) )
@ ^ [List: list @ B] :
( List
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_5319_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [List: list @ A] :
( List
= ( nil @ A ) )
@ ^ [List: list @ B] :
( List
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_5320_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_5321_antisym__singleton,axiom,
! [A: $tType,X: product_prod @ A @ A] : ( antisym @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_5322_antisym__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( antisym @ A @ R3 )
=> ( antisym @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% antisym_Restr
thf(fact_5323_product__lists__set,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) )
= ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X3: A,Ys2: list @ A] : ( member @ A @ X3 @ ( set2 @ A @ Ys2 ) )
@ Xs2
@ Xss ) ) ) ).
% product_lists_set
thf(fact_5324_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P2: A > B > $o,Xs2: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P2 @ X3 ) ) ) ) ) ).
% list_all2_iff
thf(fact_5325_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A4: A > B > $o,B3: C > D > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B3 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B3 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_5326_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_5327_inter__coset__fold,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( coset @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A4 ) ) ).
% inter_coset_fold
thf(fact_5328_congruent__def,axiom,
! [B: $tType,A: $tType] :
( ( equiv_congruent @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),F: A > B] :
! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z5: A] :
( ( F @ Y3 )
= ( F @ Z5 ) )
@ X3 ) ) ) ) ).
% congruent_def
thf(fact_5329_prod__list_OCons,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X @ Xs ) )
= ( times_times @ A @ X @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list.Cons
thf(fact_5330_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_5331_prod__list_Oappend,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( append @ A @ Xs @ Ys ) )
= ( times_times @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ ( groups5270119922927024881d_list @ A @ Ys ) ) ) ) ).
% prod_list.append
thf(fact_5332_congruentD,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F2: A > B,Y: A,Z2: A] :
( ( equiv_congruent @ A @ B @ R3 @ F2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( ( F2 @ Y )
= ( F2 @ Z2 ) ) ) ) ).
% congruentD
thf(fact_5333_congruentI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F2: A > B] :
( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( ( F2 @ Y2 )
= ( F2 @ Z3 ) ) )
=> ( equiv_congruent @ A @ B @ R3 @ F2 ) ) ).
% congruentI
thf(fact_5334_UNIV__coset,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( coset @ A @ ( nil @ A ) ) ) ).
% UNIV_coset
thf(fact_5335_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs2: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_5336_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_5337_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A4 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ A4 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_5338_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs2: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs2 @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_5339_minus__coset__filter,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_5340_congruent2__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( equiv_congruent2 @ A @ B @ C )
= ( ^ [R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F: A > B > C] :
! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R12 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y13: A,Z1: A] :
! [Y3: product_prod @ B @ B] :
( ( member @ ( product_prod @ B @ B ) @ Y3 @ R23 )
=> ( product_case_prod @ B @ B @ $o
@ ^ [Y24: B,Z22: B] :
( ( F @ Y13 @ Y24 )
= ( F @ Z1 @ Z22 ) )
@ Y3 ) )
@ X3 ) ) ) ) ).
% congruent2_def
thf(fact_5341_properties__for__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Ys: list @ B,Xs: list @ B,F2: B > A] :
( ( ( mset @ B @ Ys )
= ( mset @ B @ Xs ) )
=> ( ! [K2: B] :
( ( member @ B @ K2 @ ( set2 @ B @ Ys ) )
=> ( ( filter2 @ B
@ ^ [X3: B] :
( ( F2 @ K2 )
= ( F2 @ X3 ) )
@ Ys )
= ( filter2 @ B
@ ^ [X3: B] :
( ( F2 @ K2 )
= ( F2 @ X3 ) )
@ Xs ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F2 @ Xs )
= Ys ) ) ) ) ) ).
% properties_for_sort_key
thf(fact_5342_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A6: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A6 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_5343_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_fin.empty
thf(fact_5344_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_5345_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_5346_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs1 ) ) @ ( mset @ A @ Xs22 ) ) ) ).
% mset_mergesort_by_rel_split
thf(fact_5347_surj__mset,axiom,
! [A: $tType] :
( ( image2 @ ( list @ A ) @ ( multiset @ A ) @ ( mset @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( top_top @ ( set @ ( multiset @ A ) ) ) ) ).
% surj_mset
thf(fact_5348_mset__eq__finite,axiom,
! [A: $tType,Xs: list @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( mset @ A @ Ys2 )
= ( mset @ A @ Xs ) ) ) ) ).
% mset_eq_finite
thf(fact_5349_mset__eq__length__filter,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ Z2 )
@ Xs ) )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ Z2 )
@ Ys ) ) ) ) ).
% mset_eq_length_filter
thf(fact_5350_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),F2: A > B > C] :
( ! [Y12: A,Z12: A,Y23: B,Z23: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z12 ) @ R13 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z23 ) @ R24 )
=> ( ( F2 @ Y12 @ Y23 )
= ( F2 @ Z12 @ Z23 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R13 @ R24 @ F2 ) ) ).
% congruent2I'
thf(fact_5351_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),F2: A > B > C,Y1: A,Z13: A,Y22: B,Z24: B] :
( ( equiv_congruent2 @ A @ B @ C @ R13 @ R24 @ F2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z13 ) @ R13 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z24 ) @ R24 )
=> ( ( F2 @ Y1 @ Y22 )
= ( F2 @ Z13 @ Z24 ) ) ) ) ) ).
% congruent2D
thf(fact_5352_Gcd__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( semiring_gcd_Gcd_fin @ A @ B3 ) ) ) ) ).
% Gcd_fin.union
thf(fact_5353_properties__for__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ys: list @ A,Xs: list @ A] :
( ( ( mset @ A @ Ys )
= ( mset @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs )
= Ys ) ) ) ) ).
% properties_for_sort
thf(fact_5354_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_5355_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_5356_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_5357_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,J: nat,X: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys ) ) ) ) )
= ( add_mset @ ( 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
thf(fact_5358_sorted__list__of__multiset__mset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord6283353356039996273ltiset @ A @ ( mset @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs ) ) ) ).
% sorted_list_of_multiset_mset
thf(fact_5359_image__mset__map__of,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( image_mset @ A @ B
@ ^ [I3: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I3 ) )
@ ( mset @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
= ( mset @ B @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) ) ) ) ).
% image_mset_map_of
thf(fact_5360_image__mset_Oidentity,axiom,
! [A: $tType] :
( ( image_mset @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( multiset @ A ) ) ) ).
% image_mset.identity
thf(fact_5361_Multiset_Omset__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( mset @ A
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) )
= ( add_mset @ A @ X @ ( mset @ A @ Xs ) ) ) ) ).
% Multiset.mset_insort
thf(fact_5362_sorted__list__of__multiset__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,M4: multiset @ A] :
( ( linord6283353356039996273ltiset @ A @ ( add_mset @ A @ X @ M4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord6283353356039996273ltiset @ A @ M4 ) ) ) ) ).
% sorted_list_of_multiset_insert
thf(fact_5363_multiset_Oinj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( multiset @ A ) @ ( multiset @ B ) @ ( image_mset @ A @ B @ F2 ) @ ( top_top @ ( set @ ( multiset @ A ) ) ) ) ) ).
% multiset.inj_map
thf(fact_5364_multiset_Omap__ident,axiom,
! [A: $tType,T4: multiset @ A] :
( ( image_mset @ A @ A
@ ^ [X3: A] : X3
@ T4 )
= T4 ) ).
% multiset.map_ident
thf(fact_5365_nat__to__rat__surj__def,axiom,
( nat_to_rat_surj
= ( ^ [N2: nat] :
( product_case_prod @ nat @ nat @ rat
@ ^ [A5: nat,B4: nat] : ( fract @ ( nat_int_decode @ A5 ) @ ( nat_int_decode @ B4 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ).
% nat_to_rat_surj_def
thf(fact_5366_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 ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append2
thf(fact_5367_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P2: A > A > $o,A6: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 )
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
& ( P2 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_5368_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_5369_surj__prod__decode,axiom,
( ( image2 @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% surj_prod_decode
thf(fact_5370_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A8: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A8 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_5371_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R4 ) ) ) ).
% irrefl_def
thf(fact_5372_lexl__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R3 ) ) ) ).
% lexl_not_refl
thf(fact_5373_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B4: A] : A5 != B4
@ X3 ) ) ) ) ).
% irrefl_distinct
thf(fact_5374_list__decode_Oelims,axiom,
! [X: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_5375_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord3483353639454293061rt_key @ B @ A )
= ( ^ [Sk: ( B > A ) > ( list @ B ) > ( list @ B )] :
! [F: B > A,Xs2: list @ B,K4: A] :
( ( filter2 @ B
@ ^ [Y3: B] :
( ( F @ Y3 )
= K4 )
@ ( Sk @ F @ Xs2 ) )
= ( filter2 @ B
@ ^ [Y3: B] :
( ( F @ Y3 )
= K4 )
@ Xs2 ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_5376_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I4: set @ A,F5: A > ( set @ B ),F2: B > C,G5: D > ( set @ C ),J5: set @ D] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I2 ) @ ( F5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J2 ) @ ( F5 @ I2 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F2
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G5 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I3: A] : ( principal @ B @ ( F5 @ I3 ) )
@ I4 ) ) )
= ( ! [X3: D] :
( ( member @ D @ X3 @ J5 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ I4 )
& ! [Z5: B] :
( ( member @ B @ Z5 @ ( F5 @ Y3 ) )
=> ( member @ C @ ( F2 @ Z5 ) @ ( G5 @ X3 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_5377_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G2: A > B,F33: filter @ B,F24: filter @ A,F2: C > A,F13: filter @ C] :
( ( filterlim @ A @ B @ G2 @ F33 @ F24 )
=> ( ( filterlim @ C @ A @ F2 @ F24 @ F13 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( G2 @ ( F2 @ X3 ) )
@ F33
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_5378_filterlim__ident,axiom,
! [A: $tType,F5: filter @ A] :
( filterlim @ A @ A
@ ^ [X3: A] : X3
@ F5
@ F5 ) ).
% filterlim_ident
thf(fact_5379_filterlim__sup,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ B,F13: filter @ A,F24: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F5 @ F13 )
=> ( ( filterlim @ A @ B @ F2 @ F5 @ F24 )
=> ( filterlim @ A @ B @ F2 @ F5 @ ( sup_sup @ ( filter @ A ) @ F13 @ F24 ) ) ) ) ).
% filterlim_sup
thf(fact_5380_filterlim__inf,axiom,
! [B: $tType,A: $tType,F2: A > B,F24: filter @ B,F33: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( inf_inf @ ( filter @ B ) @ F24 @ F33 ) @ F13 )
= ( ( filterlim @ A @ B @ F2 @ F24 @ F13 )
& ( filterlim @ A @ B @ F2 @ F33 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_5381_surj__list__decode,axiom,
( ( image2 @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% surj_list_decode
thf(fact_5382_filterlim__top,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ A] : ( filterlim @ A @ B @ F2 @ ( top_top @ ( filter @ B ) ) @ F5 ) ).
% filterlim_top
thf(fact_5383_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G5: C > ( filter @ B ),B3: set @ C,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G5 @ B3 ) ) @ F5 )
= ( ! [X3: C] :
( ( member @ C @ X3 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( G5 @ X3 ) @ F5 ) ) ) ) ).
% filterlim_INF
thf(fact_5384_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A4: set @ A,F2: B > C,F5: filter @ C,G5: A > ( filter @ B )] :
( ( member @ A @ X @ A4 )
=> ( ( filterlim @ B @ C @ F2 @ F5 @ ( G5 @ X ) )
=> ( filterlim @ B @ C @ F2 @ F5 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G5 @ A4 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_5385_filterlim__If,axiom,
! [B: $tType,A: $tType,F2: A > B,G5: filter @ B,F5: filter @ A,P: A > $o,G2: A > B] :
( ( filterlim @ A @ B @ F2 @ G5 @ ( inf_inf @ ( filter @ A ) @ F5 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G2 @ G5
@ ( inf_inf @ ( filter @ A ) @ F5
@ ( principal @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( if @ B @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ G5
@ F5 ) ) ) ).
% filterlim_If
thf(fact_5386_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J5: set @ A,I: A > C,I4: set @ C,F5: C > ( set @ D ),F2: D > E,G5: A > ( set @ E )] :
( ! [M3: A,X2: B] :
( ( member @ A @ M3 @ J5 )
=> ( member @ C @ ( I @ M3 ) @ I4 ) )
=> ( ! [M3: A,X2: D] :
( ( member @ A @ M3 @ J5 )
=> ( ( member @ D @ X2 @ ( F5 @ ( I @ M3 ) ) )
=> ( member @ E @ ( F2 @ X2 ) @ ( G5 @ M3 ) ) ) )
=> ( filterlim @ D @ E @ F2
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J3: A] : ( principal @ E @ ( G5 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I3: C] : ( principal @ D @ ( F5 @ I3 ) )
@ I4 ) ) ) ) ) ).
% filterlim_base
thf(fact_5387_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_5388_list__decode_Opelims,axiom,
! [X: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( accp @ nat @ nat_list_decode_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_5389_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X3: B,Uu: list @ B] : ( cons @ A @ ( F @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_5390_zipf__zip,axiom,
! [A: $tType,B: $tType,L12: list @ A,L23: list @ B] :
( ( ( size_size @ ( list @ A ) @ L12 )
= ( size_size @ ( list @ B ) @ L23 ) )
=> ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L12 @ L23 )
= ( zip @ A @ B @ L12 @ L23 ) ) ) ).
% zipf_zip
thf(fact_5391_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F2: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B,X: A,Xs: list @ A] :
( ( F2
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( cons @ A @ X @ Xs ) )
= ( F22 @ X @ Xs @ ( F2 @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_5392_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F2: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B] :
( ( F2
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( nil @ A ) )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_5393_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ! [X5: nat,Y6: nat] :
( ( ( product_Pair @ nat @ nat @ X5 @ Y6 )
= ( nat_prod_decode @ N3 ) )
=> ( P @ Y6 ) )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_5394_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_5395_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G2: C,Ga: B > ( list @ B ) > C > C,F2: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G2 @ Ga ) @ ( map @ A @ B @ F2 ) )
= ( rec_list @ C @ A @ G2
@ ^ [X3: A,Xa4: list @ A] : ( Ga @ ( F2 @ X3 ) @ ( map @ A @ B @ F2 @ Xa4 ) ) ) ) ).
% list.rec_o_map
thf(fact_5396_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( nil @ C ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A8: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A8 @ As4 ) )
=> ! [B7: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B7 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X @ A8 @ B7 ) @ ( zipf @ A @ B @ C @ X @ As4 @ Bs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ ( cons @ B @ B7 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V3 @ Va ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% zipf.pelims
thf(fact_5397_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X3: A,Uu: list @ A] : ( insert2 @ A @ X3 ) ) ) ).
% set_rec
thf(fact_5398_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A8: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A8 @ As4 ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As4 @ ( cons @ B @ ( X @ A8 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_5399_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_5400_the__dflt__None__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( dflt_None_set @ A @ S )
= ( some @ ( set @ A ) @ S ) ) ) ).
% the_dflt_None_nonempty
thf(fact_5401_curr__surj,axiom,
! [C: $tType,B: $tType,A: $tType,G2: A > B > C,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( A > B > C ) @ G2 @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B3 @ C3 ) ) )
=> ? [X2: ( product_prod @ A @ B ) > C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ X2
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ C3 ) )
& ( ( bNF_Wellorder_curr @ A @ B @ C @ A4 @ X2 )
= G2 ) ) ) ).
% curr_surj
thf(fact_5402_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A6: set @ A,F: ( product_prod @ A @ B ) > C,A5: A] :
( if @ ( B > C ) @ ( member @ A @ A5 @ A6 )
@ ^ [B4: B] : ( F @ ( product_Pair @ A @ B @ A5 @ B4 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_5403_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S8: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S8
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S8 ) ) ) ) ).
% dflt_None_set_def
thf(fact_5404_curr__inj,axiom,
! [C: $tType,B: $tType,A: $tType,F1: ( product_prod @ A @ B ) > C,A4: set @ A,B3: set @ B,C3: set @ C,F22: ( product_prod @ A @ B ) > C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ F1
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ C3 ) )
=> ( ( member @ ( ( product_prod @ A @ B ) > C ) @ F22
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ C3 ) )
=> ( ( ( bNF_Wellorder_curr @ A @ B @ C @ A4 @ F1 )
= ( bNF_Wellorder_curr @ A @ B @ C @ A4 @ F22 ) )
= ( F1 = F22 ) ) ) ) ).
% curr_inj
thf(fact_5405_curr__in,axiom,
! [A: $tType,B: $tType,C: $tType,F2: ( product_prod @ A @ B ) > C,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ F2
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ C3 ) )
=> ( member @ ( A > B > C ) @ ( bNF_Wellorder_curr @ A @ B @ C @ A4 @ F2 ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B3 @ C3 ) ) ) ) ).
% curr_in
thf(fact_5406_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
thf(fact_5407_bij__betw__curr,axiom,
! [A: $tType,B: $tType,C: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( bij_betw @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) @ ( bNF_Wellorder_curr @ A @ B @ C @ A4 )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ C3 )
@ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B3 @ C3 ) ) ) ).
% bij_betw_curr
thf(fact_5408_init__seg__of__def,axiom,
! [A: $tType] :
( ( init_seg_of @ A )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S2 )
& ! [A5: A,B4: A,C5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ S2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R4 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_5409_bij__is__inj,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_is_inj
thf(fact_5410_bij__id,axiom,
! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% bij_id
thf(fact_5411_bij__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_5412_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_betw_imp_surj
thf(fact_5413_bij__is__surj,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_is_surj
thf(fact_5414_bij__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_uminus
thf(fact_5415_bijI_H,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ! [X2: A,Y2: A] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: B] :
? [X5: A] :
( Y2
= ( F2 @ X5 ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI'
thf(fact_5416_bij__iff,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ! [X3: B] :
? [Y3: A] :
( ( ( F2 @ Y3 )
= X3 )
& ! [Z5: A] :
( ( ( F2 @ Z5 )
= X3 )
=> ( Z5 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_5417_bij__pointE,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X2: A] :
( ( Y
= ( F2 @ X2 ) )
=> ~ ! [X10: A] :
( ( Y
= ( F2 @ X10 ) )
=> ( X10 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_5418_involuntory__imp__bij,axiom,
! [A: $tType,F2: A > A] :
( ! [X2: A] :
( ( F2 @ ( F2 @ X2 ) )
= X2 )
=> ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% involuntory_imp_bij
thf(fact_5419_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ ( bot_bot @ ( set @ B ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_5420_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) @ A4 )
=> ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_5421_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H3: B > C,S: set @ B,T2: set @ C,G2: C > A] :
( ( bij_betw @ B @ C @ H3 @ S @ T2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( G2 @ ( H3 @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T2 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_5422_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H3: B > C,S: set @ B,T2: set @ C,G2: C > A] :
( ( bij_betw @ B @ C @ H3 @ S @ T2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G2 @ ( H3 @ X3 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G2 @ T2 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_5423_bij__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: B > C] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% bij_comp
thf(fact_5424_trans__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),T4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ T4 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ T4 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_5425_antisym__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ R3 ) @ ( init_seg_of @ A ) )
=> ( R3 = S3 ) ) ) ).
% antisym_init_seg_of
thf(fact_5426_refl__on__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( init_seg_of @ A ) ) ).
% refl_on_init_seg_of
thf(fact_5427_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A4: set @ A,F2: A > B,A17: set @ B] :
( ~ ( member @ A @ B2 @ A4 )
=> ( ~ ( member @ B @ ( F2 @ B2 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F2 @ A4 @ A17 )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert2 @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_5428_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A4: set @ A,F2: A > B,A17: set @ B] :
( ~ ( member @ A @ B2 @ A4 )
=> ( ~ ( member @ B @ ( F2 @ B2 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F2 @ A4 @ A17 )
= ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert2 @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_5429_bijI,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI
thf(fact_5430_bij__def,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
& ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ) ).
% bij_def
thf(fact_5431_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B3: set @ B,C3: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B3 )
=> ( ( bij_betw @ A @ B @ F2 @ C3 @ D4 )
=> ( ( ( inf_inf @ ( set @ B ) @ B3 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D4 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_5432_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,C3: set @ A,B3: set @ B,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D4 ) )
=> ( ( bij_betw @ A @ B @ F2 @ C3 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ C3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B3 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ A4 @ B3 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_5433_o__bij,axiom,
! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
( ( ( comp @ B @ A @ A @ G2 @ F2 )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ F2 @ G2 )
= ( id @ B ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% o_bij
thf(fact_5434_finite__vimage__iff,axiom,
! [A: $tType,B: $tType,H3: A > B,F5: set @ B] :
( ( bij_betw @ A @ B @ H3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H3 @ F5 ) )
= ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimage_iff
thf(fact_5435_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_5436_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,C3: set @ B,G2: A > B,B3: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ C3 )
=> ( ( bij_betw @ A @ B @ G2 @ B3 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C3 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B3 )
@ ( sup_sup @ ( set @ B ) @ C3 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_5437_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),F2: B > C,A17: A > ( set @ C )] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ( member @ A @ J2 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A4 @ I2 ) @ ( A4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A4 @ J2 ) @ ( A4 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( bij_betw @ B @ C @ F2 @ ( A4 @ I2 ) @ ( A17 @ I2 ) ) )
=> ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A17 @ I4 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_5438_infinite__imp__bij__betw2,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [H: A > A] : ( bij_betw @ A @ A @ H @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_5439_infinite__imp__bij__betw,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [H: A > A] : ( bij_betw @ A @ A @ H @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_5440_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,B3: set @ B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ A4 )
= ( ord_less_eq @ ( set @ B ) @ B3 @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ).
% vimage_subset_eq
thf(fact_5441_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( comple5582772986160207858norder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( F2 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ).
% mono_bij_Inf
thf(fact_5442_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T5: set @ C,H3: B > C,S: set @ B,T2: set @ C,G2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H3 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) @ ( minus_minus @ ( set @ C ) @ T2 @ T5 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S4 )
=> ( ( G2 @ ( H3 @ A8 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( G2 @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( G2 @ ( H3 @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T2 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_5443_initial__segment__of__Diff,axiom,
! [A: $tType,P4: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ Q4 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ S3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q4 @ S3 ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_5444_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T5: set @ C,H3: B > C,S: set @ B,T2: set @ C,G2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H3 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) @ ( minus_minus @ ( set @ C ) @ T2 @ T5 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S4 )
=> ( ( G2 @ ( H3 @ A8 ) )
= ( one_one @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( G2 @ B7 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G2 @ ( H3 @ X3 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G2 @ T2 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_5445_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,B3: C > ( set @ A ),A4: set @ C] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X3: C] : ( image2 @ A @ B @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) ) ) ).
% bij_image_INT
thf(fact_5446_Chains__init__seg__of__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) ),R3: 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 ) ) @ R3 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_init_seg_of_Union
thf(fact_5447_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ( inj_on @ A @ B @ F2 @ ( collect @ A @ P ) )
=> ( ( P @ A3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ ( F2 @ Y2 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F2 @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_5448_sorted__list__of__multiset__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord6283353356039996273ltiset @ A )
= ( fold_mset @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_multiset_def
thf(fact_5449_bij__int__decode,axiom,
bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% bij_int_decode
thf(fact_5450_bij__enumerate,axiom,
! [S: set @ nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( bij_betw @ nat @ nat @ ( infini527867602293511546merate @ nat @ S ) @ ( top_top @ ( set @ nat ) ) @ S ) ) ).
% bij_enumerate
thf(fact_5451_bij__int__encode,axiom,
bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_int_encode
thf(fact_5452_bij__prod__decode,axiom,
bij_betw @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% bij_prod_decode
thf(fact_5453_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M4: set @ A] :
( ( finite_finite2 @ A @ M4 )
=> ? [H: nat > A] : ( bij_betw @ nat @ A @ H @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_5454_bij__list__decode,axiom,
bij_betw @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( list @ nat ) ) ) ).
% bij_list_decode
thf(fact_5455_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C7: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ C7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_5456_Chains__inits__DiffI,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) ),S3: 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 @ ( set @ ( product_prod @ A @ A ) ) )
@ ( collect @ ( set @ ( product_prod @ A @ A ) )
@ ^ [Uu: set @ ( product_prod @ A @ A )] :
? [R4: set @ ( product_prod @ A @ A )] :
( ( Uu
= ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S3 ) )
& ( member @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) ) )
@ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_inits_DiffI
thf(fact_5457_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F: B > A,S8: set @ B] :
( lattices_ord_arg_min @ B @ A @ F
@ ^ [X3: B] : ( member @ B @ X3 @ S8 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_5458_Chains__subset_H,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) )
@ ( chains @ A @ R3 ) ) ) ).
% Chains_subset'
thf(fact_5459_Chains__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R3 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ).
% Chains_subset
thf(fact_5460_Chains__alt__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ( chains @ A @ R3 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_5461_pred__on_Ochain__empty,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o] : ( pred_chain @ A @ A4 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_5462_subset_Ochain__total,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ X @ C3 )
=> ( ( member @ ( set @ A ) @ Y @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_5463_pred__on_Ochain__total,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C3: set @ A,X: A,Y: A] :
( ( pred_chain @ A @ A4 @ P @ C3 )
=> ( ( member @ A @ X @ C3 )
=> ( ( member @ A @ Y @ C3 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X
@ Y )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y
@ X ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_5464_subset_Ochain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
& ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C3 )
=> ! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X3
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y3
@ X3 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_5465_subset_OchainI,axiom,
! [A: $tType,C3: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
=> ( ! [X2: set @ A,Y2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
=> ( ( member @ ( set @ A ) @ Y2 @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X2
@ Y2 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y2
@ X2 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ) ).
% subset.chainI
thf(fact_5466_subset_Ochain__empty,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_5467_pred__on_OchainI,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A4 )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ C3 )
=> ( ( member @ A @ Y2 @ C3 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X2
@ Y2 )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y2
@ X2 ) ) ) )
=> ( pred_chain @ A @ A4 @ P @ C3 ) ) ) ).
% pred_on.chainI
thf(fact_5468_pred__on_Ochain__def,axiom,
! [A: $tType] :
( ( pred_chain @ A )
= ( ^ [A6: set @ A,P2: A > A > $o,C7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C7 @ A6 )
& ! [X3: A] :
( ( member @ A @ X3 @ C7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C7 )
=> ( ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3
@ Y3 )
| ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y3
@ X3 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_5469_chains__alt__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A6: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_5470_chain__subset__alt__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% chain_subset_alt_def
thf(fact_5471_subset__Zorn__nonempty,axiom,
! [A: $tType,A19: set @ ( set @ A )] :
( ( A19
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C9: set @ ( set @ A )] :
( ( C9
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A19 @ ( ord_less @ ( set @ A ) ) @ C9 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) @ A19 ) ) )
=> ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A19 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_5472_Union__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A19: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A19 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Union_in_chain
thf(fact_5473_Inter__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A19: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A19 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Inter_in_chain
thf(fact_5474_subset_Ochain__extend,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),Z2: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ Z2 @ A4 )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X2
@ Z2 ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_5475_pred__on_Ochain__extend,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C3: set @ A,Z2: A] :
( ( pred_chain @ A @ A4 @ P @ C3 )
=> ( ( member @ A @ Z2 @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ C3 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X2
@ Z2 ) )
=> ( pred_chain @ A @ A4 @ P @ ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) @ C3 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_5476_finite__subset__Union__chain,axiom,
! [A: $tType,A4: set @ A,B11: set @ ( set @ A ),A19: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A19 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ~ ! [B10: set @ A] :
( ( member @ ( set @ A ) @ B10 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B10 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_5477_dom__mmupd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K5: set @ A,V: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M @ K5 @ V ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M ) @ K5 ) ) ).
% dom_mmupd
thf(fact_5478_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_5479_Rep__unit,axiom,
! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_5480_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),V: B] :
( ( map_mmupd @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) @ V )
= M ) ).
% map_mmupd_empty
thf(fact_5481_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_5482_measures__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_5483_measures__lesseq,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_5484_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ! [X2: product_unit] : ( P @ ( product_Rep_unit @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_5485_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ~ ! [X2: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X2 ) ) ) ) ).
% Rep_unit_cases
thf(fact_5486_type__definition__unit,axiom,
type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% type_definition_unit
thf(fact_5487_Abs__unit__inverse,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
= Y ) ) ).
% Abs_unit_inverse
thf(fact_5488_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( ( member @ $o @ X @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( product_Abs_unit @ X )
= ( product_Abs_unit @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_unit_inject
thf(fact_5489_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X
= ( product_Abs_unit @ Y2 ) )
=> ~ ( member @ $o @ Y2 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_5490_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y2: $o] :
( ( member @ $o @ Y2 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P @ ( product_Abs_unit @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_5491_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X3: A,N2: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) @ ( power_power @ A @ X3 @ ( nat2 @ N2 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_5492_mset__set__Union,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A4 ) @ ( mset_set @ A @ B3 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_5493_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_5494_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
thf(fact_5495_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( one_one @ int ) )
= Y ) ) ).
% power_int_1_right
thf(fact_5496_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5497_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5498_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X: B] :
( ( power_int @ B @ X @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_5499_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ).
% abs_power_int_minus
thf(fact_5500_mset__set_Oempty,axiom,
! [A: $tType] :
( ( mset_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset_set.empty
thf(fact_5501_sum__multiset__singleton,axiom,
! [A: $tType,A4: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ ( multiset @ A )
@ ^ [N2: A] : ( add_mset @ A @ N2 @ ( zero_zero @ ( multiset @ A ) ) )
@ A4 )
= ( mset_set @ A @ A4 ) ) ).
% sum_multiset_singleton
thf(fact_5502_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_5503_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int,B2: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ B2 ) )
= B2 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_5504_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_5505_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_5506_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y ) ) ) ).
% power_int_minus1_right
thf(fact_5507_power__int__minus__left__odd,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_5508_power__int__minus__left__even,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) ) ) ).
% power_int_minus_left_even
thf(fact_5509_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_5510_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_commutes
thf(fact_5511_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
thf(fact_5512_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ X @ ( uminus_uminus @ int @ N ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus
thf(fact_5513_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( ( M
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( one_one @ A ) ) )
& ( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_5514_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less_eq @ int @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N4 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5515_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less @ int @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N4 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5516_power__int__minus__one__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% power_int_minus_one_minus
thf(fact_5517_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: int,B2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_5518_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
thf(fact_5519_mset__set__empty__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( mset_set @ A @ A4 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% mset_set_empty_iff
thf(fact_5520_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less @ int @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N4 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5521_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_5522_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_5523_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_5524_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A3: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_5525_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less_eq @ int @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N4
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N4 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5526_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_5527_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5528_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5529_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_5530_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_5531_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_5532_power__int__minus__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_minus_left
thf(fact_5533_Zorns__po__lemma,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ ( chains @ A @ R3 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R3 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X5 ) @ R3 ) ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R3 )
=> ( Xa2 = X2 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_5534_Partial__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_7125193373082350890der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Partial_order_Restr
thf(fact_5535_mset__set_Oinsert__remove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( mset_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% mset_set.insert_remove
thf(fact_5536_mset__set_Oremove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( mset_set @ A @ A4 )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_5537_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M: num,N: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_5538_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A18: set @ A,R13: set @ ( product_prod @ A @ A ),A25: set @ B,R24: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A18 @ R13 )
=> ( ( equiv_equiv @ B @ A25 @ R24 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R13 @ R24 @ F2 )
=> ( ( member @ A @ A1 @ A18 )
=> ( ( member @ B @ A22 @ A25 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X12 ) @ ( image @ B @ B @ R24 @ ( insert2 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R13 @ ( insert2 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_5539_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A18: set @ A,R13: set @ ( product_prod @ A @ A ),A25: set @ B,R24: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A18 @ R13 )
=> ( ( equiv_equiv @ B @ A25 @ R24 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R13 @ R24 @ F2 )
=> ( ( member @ B @ A3 @ A25 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R13
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X12 ) @ ( image @ B @ B @ R24 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_5540_quotient__eqI,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A,Y4: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ A @ Y @ Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X7 = Y4 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_5541_quotient__eq__iff,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A,Y4: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ A @ Y @ Y4 )
=> ( ( X7 = Y4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_5542_in__quotient__imp__closed,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ X7 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_5543_in__quotient__imp__non__empty,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( X7
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_5544_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > A > B] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ! [Y2: A,Z3: A] :
( ( member @ A @ Y2 @ A4 )
=> ( ( member @ A @ Z3 @ A4 )
=> ( ( F2 @ Y2 @ Z3 )
= ( F2 @ Z3 @ Y2 ) ) ) )
=> ( ! [Y2: A,Z3: A,W: A] :
( ( member @ A @ W @ A4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( ( F2 @ W @ Y2 )
= ( F2 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R3 @ R3 @ F2 ) ) ) ) ).
% congruent2_commuteI
thf(fact_5545_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A18: set @ A,R13: set @ ( product_prod @ A @ A ),A25: set @ B,R24: set @ ( product_prod @ B @ B ),F2: A > B > C] :
( ( equiv_equiv @ A @ A18 @ R13 )
=> ( ( equiv_equiv @ B @ A25 @ R24 )
=> ( ! [Y2: A,Z3: A,W: B] :
( ( member @ B @ W @ A25 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R13 )
=> ( ( F2 @ Y2 @ W )
= ( F2 @ Z3 @ W ) ) ) )
=> ( ! [Y2: B,Z3: B,W: A] :
( ( member @ A @ W @ A18 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z3 ) @ R24 )
=> ( ( F2 @ W @ Y2 )
= ( F2 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R13 @ R24 @ F2 ) ) ) ) ) ).
% congruent2I
thf(fact_5546_equiv__type,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ).
% equiv_type
thf(fact_5547_equiv__class__self,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ A @ A3 @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_5548_quotient__disj,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A,Y4: set @ A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( X7 = Y4 )
| ( ( inf_inf @ ( set @ A ) @ X7 @ Y4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_5549_UN__equiv__class__type,axiom,
! [A: $tType,B: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),X7: set @ A,B3: set @ ( set @ B )] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ ( set @ B ) @ ( F2 @ X2 ) @ B3 ) )
=> ( member @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X7 ) ) @ B3 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_5550_UN__equiv__class__type2,axiom,
! [A: $tType,B: $tType,C: $tType,A18: set @ A,R13: set @ ( product_prod @ A @ A ),A25: set @ B,R24: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),X14: set @ A,X25: set @ B,B3: set @ ( set @ C )] :
( ( equiv_equiv @ A @ A18 @ R13 )
=> ( ( equiv_equiv @ B @ A25 @ R24 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R13 @ R24 @ F2 )
=> ( ( member @ ( set @ A ) @ X14 @ ( equiv_quotient @ A @ A18 @ R13 ) )
=> ( ( member @ ( set @ B ) @ X25 @ ( equiv_quotient @ B @ A25 @ R24 ) )
=> ( ! [X13: A,X24: B] :
( ( member @ A @ X13 @ A18 )
=> ( ( member @ B @ X24 @ A25 )
=> ( member @ ( set @ C ) @ ( F2 @ X13 @ X24 ) @ B3 ) ) )
=> ( member @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X12 ) @ X25 ) )
@ X14 ) )
@ B3 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_5551_equiv__class__eq__iff,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
= ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A4 )
& ( member @ A @ Y @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_5552_eq__equiv__class__iff,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_5553_equiv__class__eq,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_5554_eq__equiv__class,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,A4: set @ A] :
( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ A @ B2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_5555_eq__equiv__class__iff2,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_5556_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_5557_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_5558_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),X7: set @ A,Y4: set @ A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F2 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X7 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ Y4 ) ) )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) ) ) )
=> ( X7 = Y4 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_5559_refines__equiv__image__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S )
=> ( ( image2 @ ( set @ A ) @ ( set @ A ) @ ( image @ A @ A @ S ) @ ( equiv_quotient @ A @ A4 @ R ) )
= ( equiv_quotient @ A @ A4 @ S ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_5560_subset__equiv__class,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_5561_equiv__class__subset,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_5562_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_5563_in__quotient__imp__in__rel,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_5564_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_5565_proj__iff,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 )
=> ( ( ( equiv_proj @ A @ A @ R3 @ X )
= ( equiv_proj @ A @ A @ R3 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% proj_iff
thf(fact_5566_disjnt__equiv__class,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_5567_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_5568_disjnt__self__iff__empty,axiom,
! [A: $tType,S: set @ A] :
( ( disjnt @ A @ S @ S )
= ( S
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjnt_self_iff_empty
thf(fact_5569_disjnt__Un1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C3 )
= ( ( disjnt @ A @ A4 @ C3 )
& ( disjnt @ A @ B3 @ C3 ) ) ) ).
% disjnt_Un1
thf(fact_5570_disjnt__Un2,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B3: set @ A] :
( ( disjnt @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( disjnt @ A @ C3 @ A4 )
& ( disjnt @ A @ C3 @ B3 ) ) ) ).
% disjnt_Un2
thf(fact_5571_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C3: set @ B,B3: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C3 )
@ ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : C3 ) )
= ( ( C3
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A4 @ B3 ) ) ) ).
% disjnt_Times2_iff
thf(fact_5572_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C3: set @ A,A4: set @ B,B3: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C3
@ ^ [Uu: A] : A4 )
@ ( product_Sigma @ A @ B @ C3
@ ^ [Uu: A] : B3 ) )
= ( ( C3
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A4 @ B3 ) ) ) ).
% disjnt_Times1_iff
thf(fact_5573_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C3: A > ( set @ B ),B3: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ C3 ) @ ( product_Sigma @ A @ B @ B3 @ C3 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( ( C3 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A4 @ B3 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_5574_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( ( Y6 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R3 ) )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_5575_wo__rel_OTOTALS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R3 ) )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa2 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X5 ) @ R3 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_5576_wo__rel_Omax2__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= B2 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_5577_disjnt__empty1,axiom,
! [A: $tType,A4: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% disjnt_empty1
thf(fact_5578_disjnt__empty2,axiom,
! [A: $tType,A4: set @ A] : ( disjnt @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_5579_well__order__induct__imp,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( ( Y6 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R3 ) )
=> ( ( member @ A @ Y6 @ ( field2 @ A @ R3 ) )
=> ( P @ Y6 ) ) )
=> ( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
=> ( P @ X2 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( P @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_5580_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_5581_wo__rel_Omax2__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_5582_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= B2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_5583_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= A3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_5584_disjoint__UN__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: B > ( set @ A ),I4: set @ B] :
( ( disjnt @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B3 @ I4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( disjnt @ A @ A4 @ ( B3 @ X3 ) ) ) ) ) ).
% disjoint_UN_iff
thf(fact_5585_wo__rel_Omax2__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_5586_wo__rel_Ocases__Total,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_5587_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_5588_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_5589_card__Un__disjnt,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( disjnt @ A @ A4 @ B3 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_5590_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),X3: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_5591_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R3 @ B3 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_5592_sum__card__image,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pairwise @ A
@ ^ [S2: A,T3: A] : ( disjnt @ B @ ( F2 @ S2 ) @ ( F2 @ T3 ) )
@ A4 )
=> ( ( groups7311177749621191930dd_sum @ ( set @ B ) @ nat @ ( finite_card @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A5: A] : ( finite_card @ B @ ( F2 @ A5 ) )
@ A4 ) ) ) ) ).
% sum_card_image
thf(fact_5593_wo__rel_Ominim__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B3 ) @ B3 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_5594_pairwise__image,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,F2: B > A,S3: set @ B] :
( ( pairwise @ A @ R3 @ ( image2 @ B @ A @ F2 @ S3 ) )
= ( pairwise @ B
@ ^ [X3: B,Y3: B] :
( ( ( F2 @ X3 )
!= ( F2 @ Y3 ) )
=> ( R3 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
@ S3 ) ) ).
% pairwise_image
thf(fact_5595_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_5596_pairwise__trivial,axiom,
! [A: $tType,I4: set @ A] :
( pairwise @ A
@ ^ [I3: A,J3: A] : J3 != I3
@ I4 ) ).
% pairwise_trivial
thf(fact_5597_wo__rel_Ominim__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( bNF_We6954850376910717587_minim @ A @ R3 @ A4 )
= ( the @ A @ ( bNF_We4791949203932849705sMinim @ A @ R3 @ A4 ) ) ) ) ).
% wo_rel.minim_def
thf(fact_5598_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A4: A] : ( pairwise @ A @ P @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_5599_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R3 @ B3 @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B3 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_5600_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S8: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ S8 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_5601_wo__rel_OisMinim__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R3 @ A4 @ B2 )
= ( ( member @ A @ B2 @ A4 )
& ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X3 ) @ R3 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_5602_wo__rel_Oequals__minim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ B3 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B7 ) @ R3 ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R3 @ B3 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_5603_wo__rel_Ominim__least,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B3 ) @ B2 ) @ R3 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_5604_wo__rel_Ominim__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B3 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_5605_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_5606_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B3: set @ A,F5: A > ( filter @ B ),P: B > $o] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ B3 )
=> ! [B7: A] :
( ( member @ A @ B7 @ B3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B3 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A8 ) @ ( F5 @ B7 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ B3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ B3 )
& ( eventually @ B @ P @ ( F5 @ X3 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_5607_pair__lessI2,axiom,
! [A3: nat,B2: nat,S3: nat,T4: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S3 @ T4 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B2 @ T4 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_5608_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X4: A] : ( P @ X4 ) ) ) ).
% eventually_top
thf(fact_5609_eventually__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X3: A] : P
@ F5 )
= P ) ) ).
% eventually_const
thf(fact_5610_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_5611_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_5612_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
? [X4: B] : ( P @ X3 @ X4 )
@ F5 )
= ( ? [Y9: A > B] :
( eventually @ A
@ ^ [X3: A] : ( P @ X3 @ ( Y9 @ X3 ) )
@ F5 ) ) ) ).
% eventually_ex
thf(fact_5613_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: filter @ A,F2: B > A,G5: filter @ B] :
( ( eventually @ A @ P @ F5 )
=> ( ( filterlim @ B @ A @ F2 @ F5 @ G5 )
=> ( eventually @ B
@ ^ [X3: B] : ( P @ ( F2 @ X3 ) )
@ G5 ) ) ) ).
% eventually_compose_filterlim
thf(fact_5614_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F14: filter @ A,F24: filter @ B,F25: filter @ B,F2: B > A,G2: B > A] :
( ( F13 = F14 )
=> ( ( F24 = F25 )
=> ( ( eventually @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= ( G2 @ X3 ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F2 @ F13 @ F24 )
= ( filterlim @ B @ A @ G2 @ F14 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_5615_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F: A > B,F26: filter @ B,F15: filter @ A] :
! [P2: B > $o] :
( ( eventually @ B @ P2 @ F26 )
=> ( eventually @ A
@ ^ [X3: A] : ( P2 @ ( F @ X3 ) )
@ F15 ) ) ) ) ).
% filterlim_iff
thf(fact_5616_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y2: B] :
( eventually @ A
@ ^ [X3: A] : ( P @ X3 @ Y2 )
@ Net )
=> ( eventually @ A
@ ^ [X3: A] :
! [X4: B] : ( P @ X3 @ X4 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_5617_eventually__inf,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,F9: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F5 @ F9 ) )
= ( ? [Q: A > $o,R2: A > $o] :
( ( eventually @ A @ Q @ F5 )
& ( eventually @ A @ R2 @ F9 )
& ! [X3: A] :
( ( ( Q @ X3 )
& ( R2 @ X3 ) )
=> ( P @ X3 ) ) ) ) ) ).
% eventually_inf
thf(fact_5618_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X3: A] : X3 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_5619_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( C3
=> ( P @ X3 ) )
@ F5 )
= ( C3
=> ( eventually @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_5620_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( C3
| ( P @ X3 ) )
@ F5 )
= ( C3
| ( eventually @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_5621_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
| C3 )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
| C3 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_5622_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ F5 )
=> ( ( eventually @ A @ P @ F5 )
=> ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_mp
thf(fact_5623_eventually__True,axiom,
! [A: $tType,F5: filter @ A] :
( eventually @ A
@ ^ [X3: A] : $true
@ F5 ) ).
% eventually_True
thf(fact_5624_eventually__conj,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A @ Q2 @ F5 )
=> ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) )
@ F5 ) ) ) ).
% eventually_conj
thf(fact_5625_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A @ Q2 @ F5 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( ( Q2 @ I2 )
=> ( R @ I2 ) ) )
=> ( eventually @ A @ R @ F5 ) ) ) ) ).
% eventually_elim2
thf(fact_5626_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [N2: A] :
( ( P @ N2 )
= ( Q2 @ N2 ) )
@ F5 )
=> ( ( eventually @ A @ P @ F5 )
= ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_subst
thf(fact_5627_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ F5 )
=> ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_rev_mp
thf(fact_5628_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
& ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_conj_iff
thf(fact_5629_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ~ ( eventually @ A @ Q2 @ F5 )
=> ~ ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ F5 ) ) ) ).
% not_eventually_impI
thf(fact_5630_eventually__happens_H,axiom,
! [A: $tType,F5: filter @ A,P: A > $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F5 )
=> ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_5631_eventually__happens,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ( eventually @ A @ P @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_5632_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_5633_trivial__limit__def,axiom,
! [A: $tType,F5: filter @ A] :
( ( F5
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X3: A] : $false
@ F5 ) ) ).
% trivial_limit_def
thf(fact_5634_eventually__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] : P
@ F5 )
= ( P
| ( F5
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_5635_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X2: A] :
~ ( P @ X2 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_5636_eventually__sup,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,F9: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F5 @ F9 ) )
= ( ( eventually @ A @ P @ F5 )
& ( eventually @ A @ P @ F9 ) ) ) ).
% eventually_sup
thf(fact_5637_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X2 )
@ Net ) )
=> ( eventually @ B
@ ^ [X3: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( P @ X3 @ Y3 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5638_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B
@ ^ [X3: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( P @ X3 @ Y3 ) )
@ Net )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X3 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5639_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5640_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5641_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ B,G5: filter @ A,F9: filter @ B,G6: filter @ A,F10: A > B] :
( ( filterlim @ A @ B @ F2 @ F5 @ G5 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F5 @ F9 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G6 @ G5 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= ( F10 @ X3 ) )
@ G6 )
=> ( filterlim @ A @ B @ F10 @ F9 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_5642_filterlim__principal,axiom,
! [B: $tType,A: $tType,F2: A > B,S: set @ B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( principal @ B @ S ) @ F5 )
= ( eventually @ A
@ ^ [X3: A] : ( member @ B @ ( F2 @ X3 ) @ S )
@ F5 ) ) ).
% filterlim_principal
thf(fact_5643_le__principal,axiom,
! [A: $tType,F5: filter @ A,A4: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ ( principal @ A @ A4 ) )
= ( eventually @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 )
@ F5 ) ) ).
% le_principal
thf(fact_5644_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I4: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( member @ A @ I @ I4 )
=> ( ( eventually @ B @ P @ ( F5 @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I4 ) ) ) ) ) ).
% eventually_INF1
thf(fact_5645_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,S3: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F5 @ ( principal @ A @ S3 ) ) )
= ( eventually @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( P @ X3 ) )
@ F5 ) ) ).
% eventually_inf_principal
thf(fact_5646_eventually__Inf__base,axiom,
! [A: $tType,B3: set @ ( filter @ A ),P: A > $o] :
( ( B3
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B3 )
=> ! [G7: filter @ A] :
( ( member @ ( filter @ A ) @ G7 @ B3 )
=> ? [X5: filter @ A] :
( ( member @ ( filter @ A ) @ X5 @ B3 )
& ( ord_less_eq @ ( filter @ A ) @ X5 @ ( inf_inf @ ( filter @ A ) @ F6 @ G7 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) )
= ( ? [X3: filter @ A] :
( ( member @ ( filter @ A ) @ X3 @ B3 )
& ( eventually @ A @ P @ X3 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_5647_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ A4 ) ) )
= ( ? [Q: A > B > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( eventually @ B @ ( Q @ X3 ) @ ( F5 @ X3 ) ) )
& ! [Y3: B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( Q @ X3 @ Y3 ) )
=> ( P @ Y3 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_5648_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F2: A > ( set @ B ),A4: set @ B,F5: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F2 @ ( finite5375528669736107172at_top @ B @ A4 ) @ F5 )
= ( ! [X4: set @ B] :
( ( ( finite_finite2 @ B @ X4 )
& ( ord_less_eq @ ( set @ B ) @ X4 @ A4 ) )
=> ( eventually @ A
@ ^ [Y3: A] :
( ( finite_finite2 @ B @ ( F2 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ X4 @ ( F2 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ Y3 ) @ A4 ) )
@ F5 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_5649_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ Z10 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5650_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ Z10 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5651_butlast__upt,axiom,
! [M: nat,N: nat] :
( ( butlast @ nat @ ( upt @ M @ N ) )
= ( upt @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_5652_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ ( F2 @ X3 ) @ Z10 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5653_pair__lessI1,axiom,
! [A3: nat,B2: nat,S3: nat,T4: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B2 @ T4 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_5654_takeWhile__not__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( takeWhile @ A
@ ^ [Y3: A] :
( Y3
!= ( last @ A @ Xs ) )
@ Xs )
= ( butlast @ A @ Xs ) ) ) ).
% takeWhile_not_last
thf(fact_5655_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: B > ( filter @ A ),B3: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ B3 ) ) )
= ( ? [X4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X4 @ B3 )
& ( finite_finite2 @ B @ X4 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ X4 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_5656_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ Z10 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5657_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs2: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% butlast_conv_take
thf(fact_5658_hd__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( butlast @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_butlast
thf(fact_5659_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_5660_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G2: B > A,Y4: set @ B,X7: set @ A,F5: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G2 @ Y4 ) @ X7 )
=> ( ( eventually @ B
@ ^ [X3: B] : ( member @ B @ X3 @ Y4 )
@ F5 )
=> ( ( map_filter_on @ A @ C @ X7 @ F2 @ ( map_filter_on @ B @ A @ Y4 @ G2 @ F5 ) )
= ( map_filter_on @ B @ C @ Y4 @ ( comp @ A @ C @ B @ F2 @ G2 ) @ F5 ) ) ) ) ).
% map_filter_on_comp
thf(fact_5661_smin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS2 )
=> ( ( member @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_5662_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S8: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P2: A > $o] :
! [X3: filter @ A] :
( ( member @ ( filter @ A ) @ X3 @ S8 )
=> ( eventually @ A @ P2 @ X3 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_5663_smin__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] :
( ( X7
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_5664_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X4: set @ A,F: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P2: B > $o] :
( eventually @ A
@ ^ [X3: A] :
( ( P2 @ ( F @ X3 ) )
& ( member @ A @ X3 @ X4 ) )
@ F7 ) ) ) ) ).
% map_filter_on_def
thf(fact_5665_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P2: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_5666_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X7: set @ A,F5: filter @ A,P: B > $o,F2: A > B] :
( ( eventually @ A
@ ^ [X3: A] : ( member @ A @ X3 @ X7 )
@ F5 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X7 @ F2 @ F5 ) )
= ( eventually @ A
@ ^ [X3: A] :
( ( P @ ( F2 @ X3 ) )
& ( member @ A @ X3 @ X7 ) )
@ F5 ) ) ) ).
% eventually_map_filter_on
thf(fact_5667_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F11: filter @ A] :
( abs_filter @ A
@ ^ [P2: A > $o] :
( ( eventually @ A @ P2 @ F7 )
& ( eventually @ A @ P2 @ F11 ) ) ) ) ) ).
% sup_filter_def
thf(fact_5668_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S8: set @ A] : ( abs_filter @ A @ ( ball @ A @ S8 ) ) ) ) ).
% principal_def
thf(fact_5669_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P5: A > $o] :
! [X6: A] : ( P5 @ X6 ) ) ) ).
% top_filter_def
thf(fact_5670_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F11: filter @ A] :
( abs_filter @ A
@ ^ [P2: A > $o] :
? [Q: A > $o,R2: A > $o] :
( ( eventually @ A @ Q @ F7 )
& ( eventually @ A @ R2 @ F11 )
& ! [X3: A] :
( ( ( Q @ X3 )
& ( R2 @ X3 ) )
=> ( P2 @ X3 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_5671_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y4: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X7: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y4 )
=> ( ( member @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ Y4 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ X7 ) @ Y4 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_5672_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X3 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5673_pair__leqI2,axiom,
! [A3: nat,B2: nat,S3: nat,T4: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S3 @ T4 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B2 @ T4 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_5674_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_5675_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( plus_plus @ nat @ N2 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_5676_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_5677_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X3: nat] : ( F2 @ ( suc @ X3 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_5678_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N11: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N11 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_5679_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X3: A] : X3 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_5680_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% pair_leq_def
thf(fact_5681_smax__emptyI,axiom,
! [Y4: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y4 )
=> ( ( Y4
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( 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 ) ) ) @ Y4 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_5682_filterlim__atMost__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_atMost_at_top
thf(fact_5683_filterlim__lessThan__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_lessThan_at_top
thf(fact_5684_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_5685_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_5686_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X3: A] :
! [Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( P @ Y3 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5687_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X3 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_5688_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X3 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5689_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F5: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ F5 )
=> ( filterlim @ B @ A @ G2 @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5690_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ Z10 @ ( F2 @ X3 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5691_pair__leqI1,axiom,
! [A3: nat,B2: nat,S3: nat,T4: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S3 ) @ ( product_Pair @ nat @ nat @ B2 @ T4 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_5692_wmin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS2 )
=> ( ( member @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_5693_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ XS2 ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_5694_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( 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
thf(fact_5695_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_5696_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N2: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_5697_max__rpair__set,axiom,
fun_reduction_pair @ ( 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
thf(fact_5698_min__rpair__set,axiom,
fun_reduction_pair @ ( 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
thf(fact_5699_wmax__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X7 )
=> ( member @ ( 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 ) ) ) @ X7 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_5700_wmin__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_5701_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( 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
thf(fact_5702_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K4 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_5703_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F: A > B,F7: filter @ B] :
( abs_filter @ A
@ ^ [P2: A > $o] :
? [Q: B > $o] :
( ( eventually @ B @ Q @ F7 )
& ! [X3: A] :
( ( Q @ ( F @ X3 ) )
=> ( P2 @ X3 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_5704_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K4 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_5705_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5706_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( filtercomap @ A @ B @ F2 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtercomap_bot
thf(fact_5707_atLeast__empty__triv,axiom,
! [A: $tType] :
( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atLeast_empty_triv
thf(fact_5708_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F5: filter @ A,F2: B > A] :
( ( eventually @ A @ P @ F5 )
=> ( eventually @ B
@ ^ [X3: B] : ( P @ ( F2 @ X3 ) )
@ ( filtercomap @ B @ A @ F2 @ F5 ) ) ) ).
% eventually_filtercomapI
thf(fact_5709_filtercomap__top,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( filtercomap @ A @ B @ F2 @ ( top_top @ ( filter @ B ) ) )
= ( top_top @ ( filter @ A ) ) ) ).
% filtercomap_top
thf(fact_5710_Sup__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
= ( top_top @ A ) ) ) ).
% Sup_atLeast
thf(fact_5711_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_5712_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_5713_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_5714_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_5715_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_5716_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ D3 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_5717_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
thf(fact_5718_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5719_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F2: A > B,F13: filter @ B,F24: filter @ B] :
( ( filtercomap @ A @ B @ F2 @ ( inf_inf @ ( filter @ B ) @ F13 @ F24 ) )
= ( inf_inf @ ( filter @ A ) @ ( filtercomap @ A @ B @ F2 @ F13 ) @ ( filtercomap @ A @ B @ F2 @ F24 ) ) ) ).
% filtercomap_inf
thf(fact_5720_filtercomap__ident,axiom,
! [A: $tType,F5: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X3: A] : X3
@ F5 )
= F5 ) ).
% filtercomap_ident
thf(fact_5721_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: B > C,F5: filter @ C] :
( ( filtercomap @ A @ B @ F2 @ ( filtercomap @ B @ C @ G2 @ F5 ) )
= ( filtercomap @ A @ C
@ ^ [X3: A] : ( G2 @ ( F2 @ X3 ) )
@ F5 ) ) ).
% filtercomap_filtercomap
thf(fact_5722_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_UNIV_eq_Ici
thf(fact_5723_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
thf(fact_5724_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_5725_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ! [P3: A > $o] :
( ( eventually @ A @ P3 @ F5 )
=> ? [X5: B] : ( P3 @ ( F2 @ X5 ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_5726_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F2: A > B,F13: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F2 @ F13 ) @ ( filtercomap @ A @ B @ F2 @ F24 ) ) @ ( filtercomap @ A @ B @ F2 @ ( sup_sup @ ( filter @ B ) @ F13 @ F24 ) ) ) ).
% filtercomap_sup
thf(fact_5727_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,F5: C > ( filter @ B ),B3: set @ C] :
( ( filtercomap @ A @ B @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B3 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B4: C] : ( filtercomap @ A @ B @ F2 @ ( F5 @ B4 ) )
@ B3 ) ) ) ).
% filtercomap_INF
thf(fact_5728_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_5729_ivl__disj__int__one_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(8)
thf(fact_5730_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_5731_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_5732_ivl__disj__int__one_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(6)
thf(fact_5733_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_5734_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert2 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_5735_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_5736_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_5737_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > C,F5: B > ( filter @ C ),B3: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B4: B] : ( filtercomap @ A @ C @ F2 @ ( F5 @ B4 ) )
@ B3 ) )
@ ( filtercomap @ A @ C @ F2 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F5 @ B3 ) ) ) ) ).
% filtercomap_SUP
thf(fact_5738_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_5739_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert2 @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5740_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_5741_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A8: A,B7: A,X2: A] :
( ( member @ A @ A8 @ S )
=> ( ( member @ A @ B7 @ S )
=> ( ( ord_less_eq @ A @ A8 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ B7 )
=> ( member @ A @ X2 @ S ) ) ) ) )
=> ? [A8: A,B7: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B7 ) )
| ( S
= ( set_ord_atMost @ A @ B7 ) )
| ( S
= ( set_ord_greaterThan @ A @ A8 ) )
| ( S
= ( set_ord_atLeast @ A @ A8 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A8 @ B7 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A8 @ B7 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A8 @ B7 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A8 @ B7 ) ) ) ) ) ).
% interval_cases
thf(fact_5742_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 ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_5743_coinduct3,axiom,
! [A: $tType,F2: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ( member @ A @ A3 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7
@ ( F2
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X3: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F2 @ X3 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) ) ) ).
% coinduct3
thf(fact_5744_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_5745_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_5746_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_5747_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_5748_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_5749_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_5750_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_5751_gfp__rolling,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [G2: A > B,F2: B > A] :
( ( order_mono @ A @ B @ G2 )
=> ( ( order_mono @ B @ A @ F2 )
=> ( ( G2
@ ( complete_lattice_gfp @ A
@ ^ [X3: A] : ( F2 @ ( G2 @ X3 ) ) ) )
= ( complete_lattice_gfp @ B
@ ^ [X3: B] : ( G2 @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_5752_gfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T4: A] :
( ( complete_lattice_gfp @ A
@ ^ [X3: A] : T4 )
= T4 ) ) ).
% gfp_const
thf(fact_5753_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A > A] :
( ! [X2: A,Y2: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 @ W ) @ ( F2 @ Y2 @ Z3 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X3: A] : ( complete_lattice_gfp @ A @ ( F2 @ X3 ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X3: A] : ( F2 @ X3 @ X3 ) ) ) ) ) ).
% gfp_gfp
thf(fact_5754_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_5755_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_5756_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
thf(fact_5757_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_mult
thf(fact_5758_gfp__fun__UnI2,axiom,
! [A: $tType,F2: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) )
=> ( member @ A @ A3 @ ( F2 @ ( sup_sup @ ( set @ A ) @ X7 @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_5759_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ U2 @ ( F @ U2 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_5760_def__Collect__coinduct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > A > $o,A3: A,X7: set @ A] :
( ( A4
= ( complete_lattice_gfp @ ( set @ A )
@ ^ [W3: set @ A] : ( collect @ A @ ( P @ W3 ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A )
@ ^ [W3: set @ A] : ( collect @ A @ ( P @ W3 ) ) )
=> ( ( member @ A @ A3 @ X7 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ X7 )
=> ( P @ ( sup_sup @ ( set @ A ) @ X7 @ A4 ) @ Z3 ) )
=> ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% def_Collect_coinduct
thf(fact_5761_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ord_less_eq @ A @ D3 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_5762_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_5763_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_5764_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_5765_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(3)
thf(fact_5766_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
thf(fact_5767_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_5768_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_5769_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_5770_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_5771_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_5772_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_5773_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(5)
thf(fact_5774_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_5775_coinduct__set,axiom,
! [A: $tType,F2: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ( member @ A @ A3 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( F2 @ ( sup_sup @ ( set @ A ) @ X7 @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) ) ) ).
% coinduct_set
thf(fact_5776_def__coinduct__set,axiom,
! [A: $tType,A4: set @ A,F2: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( A4
= ( complete_lattice_gfp @ ( set @ A ) @ F2 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ( member @ A @ A3 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( F2 @ ( sup_sup @ ( set @ A ) @ X7 @ A4 ) ) )
=> ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_5777_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X7: A,F2: A > A] :
( ( ord_less_eq @ A @ X7 @ ( F2 @ ( sup_sup @ A @ X7 @ ( complete_lattice_gfp @ A @ F2 ) ) ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X7 @ ( complete_lattice_gfp @ A @ F2 ) ) @ ( F2 @ ( sup_sup @ A @ X7 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_5778_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F2: A > A,X7: A] :
( ( A4
= ( complete_lattice_gfp @ A @ F2 ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ X7 @ ( F2 @ ( sup_sup @ A @ X7 @ A4 ) ) )
=> ( ord_less_eq @ A @ X7 @ A4 ) ) ) ) ) ).
% def_coinduct
thf(fact_5779_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,X7: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ X7 @ ( F2 @ ( sup_sup @ A @ X7 @ ( complete_lattice_gfp @ A @ F2 ) ) ) )
=> ( ord_less_eq @ A @ X7 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ).
% coinduct
thf(fact_5780_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_5781_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_5782_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5783_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_5784_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_5785_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_5786_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_5787_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_5788_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_5789_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_5790_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_5791_coinduct3__lemma,axiom,
! [A: $tType,X7: set @ A,F2: ( set @ A ) > ( set @ A )] :
( ( ord_less_eq @ ( set @ A ) @ X7
@ ( F2
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X3: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F2 @ X3 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ord_less_eq @ ( set @ A )
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X3: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F2 @ X3 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) )
@ ( F2
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X3: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F2 @ X3 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F2 ) ) ) ) ) ) ) ).
% coinduct3_lemma
thf(fact_5792_def__coinduct3,axiom,
! [A: $tType,A4: set @ A,F2: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( A4
= ( complete_lattice_gfp @ ( set @ A ) @ F2 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F2 )
=> ( ( member @ A @ A3 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7
@ ( F2
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X3: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F2 @ X3 ) @ X7 ) @ A4 ) ) ) )
=> ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% def_coinduct3
thf(fact_5793_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q7: A,R6: A] :
( ( ( euclid7384307370059645450egment @ A @ R6 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R6 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R6
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q7 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R6 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q7 @ B2 ) @ R6 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_5794_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q4: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R3 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R3 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_5795_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q4: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R3 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q4 ) ) ) ) ) ) ).
% div_eqI
thf(fact_5796_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs @ int @ ( euclid7384307370059645450egment @ int @ K ) )
= ( one_one @ int ) ) ).
% abs_division_segment
thf(fact_5797_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_5798_division__segment__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
= ( one_one @ A ) ) ) ).
% division_segment_numeral
thf(fact_5799_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_5800_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
= A3 ) ) ).
% division_segment_euclidean_size
thf(fact_5801_division__segment__nat__def,axiom,
( ( euclid7384307370059645450egment @ nat )
= ( ^ [N2: nat] : ( one_one @ nat ) ) ) ).
% division_segment_nat_def
thf(fact_5802_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_5803_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_5804_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K4: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_5805_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q4: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R3 ) @ B2 )
= Q4 ) ) ) ) ) ).
% div_bounded
thf(fact_5806_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_5807_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R4 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( B4 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B4 ) @ R4 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_5808_total__inv__image,axiom,
! [B: $tType,A: $tType,F2: A > B,R3: set @ ( product_prod @ B @ B )] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R3 )
=> ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R3 @ F2 ) ) ) ) ).
% total_inv_image
thf(fact_5809_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_5810_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R3 @ F2 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_5811_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B > A,P4: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y3: B,X3: C] : ( F2 @ X3 @ Y3 )
@ ( product_swap @ C @ B @ P4 ) )
= ( product_case_prod @ C @ B @ A @ F2 @ P4 ) ) ).
% case_swap
thf(fact_5812_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A4: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A4 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_5813_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_5814_AboveS__disjoint,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( order_AboveS @ A @ R3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_5815_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ B ),F: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) @ R4 ) ) ) ) ) ).
% inv_image_def
thf(fact_5816_product__swap,axiom,
! [B: $tType,A: $tType,A4: set @ B,B3: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : A4 ) ) ).
% product_swap
thf(fact_5817_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P6 ) @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% prod.swap_def
thf(fact_5818_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R2: set @ ( product_prod @ A @ A ),S8: set @ ( product_prod @ A @ A ),F: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R2 @ F ) @ ( inv_image @ A @ B @ S8 @ F ) ) ) ) ).
% rp_inv_image_def
thf(fact_5819_wo__rel_Osuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B2 @ B3 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R3 @ B3 )
!= B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B3 ) ) @ R3 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_5820_wo__rel_Osuc__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B3 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_5821_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R3 @ B3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B3 ) @ A3 ) @ R3 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_5822_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R3 @ B3 ) )
=> ( ! [A16: A] :
( ( member @ A @ A16 @ ( order_AboveS @ A @ R3 @ B3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A16 ) @ R3 ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R3 @ B3 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_5823_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B3 ) @ ( order_AboveS @ A @ R3 @ B3 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_5824_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R4 ) )
@ ^ [Xs2: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Xs2 ) ) ) ) ).
% lenlex_def
thf(fact_5825_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( ( order_AboveS @ A @ R3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ A4 ) ) @ R3 )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R3 @ A4 ) )
=> ( member @ A @ B2 @ A4 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_5826_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B3: C > ( set @ D ),F2: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X2: C] :
( ( member @ C @ X2 @ A4 )
=> ( ( B3 @ X2 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X3: C] : ( image2 @ D @ B @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B3 @ A4 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( B3 @ X3 ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_5827_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_5828_bdd__above__Un,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( condit941137186595557371_above @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( condit941137186595557371_above @ A @ A4 )
& ( condit941137186595557371_above @ A @ B3 ) ) ) ) ).
% bdd_above_Un
thf(fact_5829_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_5830_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F2: B > A,G2: B > A,A4: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( sup_sup @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 ) )
= ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
& ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_5831_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I4: set @ B,A4: B > ( set @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( condit941137186595557371_above @ A @ ( A4 @ X3 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_5832_bdd__above__Int2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: set @ A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ B3 )
=> ( condit941137186595557371_above @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% bdd_above_Int2
thf(fact_5833_bdd__above__Int1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ( condit941137186595557371_above @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% bdd_above_Int1
thf(fact_5834_total__less__than,axiom,
total_on @ nat @ ( top_top @ ( set @ nat ) ) @ less_than ).
% total_less_than
thf(fact_5835_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B3: set @ A,A4: set @ A] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ B7 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B3 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_5836_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S ) @ A3 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ X3 @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_5837_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Y: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X7 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X7 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ X7 )
& ( ord_less @ A @ Y @ X3 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_5838_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_5839_measures__def,axiom,
! [A: $tType] :
( ( measures @ A )
= ( ^ [Fs2: list @ ( A > nat )] :
( inv_image @ ( list @ nat ) @ A @ ( lex @ nat @ less_than )
@ ^ [A5: A] :
( map @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ A5 )
@ Fs2 ) ) ) ) ).
% measures_def
thf(fact_5840_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_5841_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G2: C > A,B3: set @ C,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G2 @ B3 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B3 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_5842_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_5843_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A3: A] :
( ( condit941137186595557371_above @ A @ X7 )
=> ( ( ( X7
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X7 ) )
= A3 ) )
& ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X7 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_5844_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A3: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X7 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X7 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_5845_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B3 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_5846_wo__rel_Oofilter__UNION,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( order_ofilter @ A @ R3 @ ( A4 @ I2 ) ) )
=> ( order_ofilter @ A @ R3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) ) ) ).
% wo_rel.ofilter_UNION
thf(fact_5847_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_5848_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,G2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ A4 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_5849_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G2: B > A,B3: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ B3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B3 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_5850_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit941137186595557371_above @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_5851_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ A @ ( F2 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_5852_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ B3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_5853_wo__rel_Oofilter__AboveS__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( order_AboveS @ A @ R3 @ A4 ) )
= ( field2 @ A @ R3 ) ) ) ) ).
% wo_rel.ofilter_AboveS_Field
thf(fact_5854_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F: A > nat,R2: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R2 )
@ ^ [X3: A] : ( product_Pair @ nat @ A @ ( F @ X3 ) @ X3 ) ) ) ) ).
% mlex_prod_def
thf(fact_5855_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( complete_Sup_Sup @ A @ S )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X3: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_5856_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A4 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F2 @ ( A4 @ X3 ) )
@ I4 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I4 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_5857_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_5858_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A6: set @ A,B5: set @ A] :
( ( order_ofilter @ A @ R4 @ A6 )
& ( A6
!= ( field2 @ A @ R4 ) )
& ( order_ofilter @ A @ R4 @ B5 )
& ( B5
!= ( field2 @ A @ R4 ) )
& ( ord_less @ ( set @ A ) @ A6 @ B5 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_5859_wo__rel_Oofilter__under__UNION,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( A4
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ A @ ( set @ A ) @ ( order_under @ A @ R3 ) @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_under_UNION
thf(fact_5860_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B3: C > ( set @ D ),F2: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X2: C] :
( ( member @ C @ X2 @ A4 )
=> ( ( B3 @ X2 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X3: C] : ( image2 @ D @ B @ F2 @ ( B3 @ X3 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B3 @ A4 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( B3 @ X3 ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_5861_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_5862_bdd__below__Un,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( condit1013018076250108175_below @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( condit1013018076250108175_below @ A @ A4 )
& ( condit1013018076250108175_below @ A @ B3 ) ) ) ) ).
% bdd_below_Un
thf(fact_5863_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F2: B > A,G2: B > A,A4: set @ B] :
( ( condit1013018076250108175_below @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( inf_inf @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 ) )
= ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
& ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_5864_bdd__below__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X7: set @ A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ X7 ) )
= ( condit941137186595557371_above @ A @ X7 ) ) ) ).
% bdd_below_uminus
thf(fact_5865_bdd__above__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X7: set @ A] :
( ( condit941137186595557371_above @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ X7 ) )
= ( condit1013018076250108175_below @ A @ X7 ) ) ) ).
% bdd_above_uminus
thf(fact_5866_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I4: set @ B,A4: B > ( set @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( condit1013018076250108175_below @ A @ ( A4 @ X3 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_5867_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_5868_bdd__below__Int1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ( condit1013018076250108175_below @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% bdd_below_Int1
thf(fact_5869_bdd__below__Int2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: set @ A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ B3 )
=> ( condit1013018076250108175_below @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% bdd_below_Int2
thf(fact_5870_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ A3 @ X3 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_5871_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B3: set @ A,A4: set @ A] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ B3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ X5 @ B7 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_5872_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Y: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X7 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X7 ) @ Y )
= ( ? [X3: A] :
( ( member @ A @ X3 @ X7 )
& ( ord_less @ A @ X3 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_5873_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_5874_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_5875_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B3: set @ B,F2: C > A,A4: set @ C,G2: B > A] :
( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F2 @ A4 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G2 @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B3 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_5876_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B3 ) @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_5877_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A3: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X7 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X7 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_5878_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A3: A] :
( ( condit1013018076250108175_below @ A @ X7 )
=> ( ( ( X7
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X7 ) )
= A3 ) )
& ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X7 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_5879_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_5880_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( ord_less @ A @ ( F2 @ X3 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_5881_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,G2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ A4 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_5882_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: C > B,A4: set @ C,A3: C] :
( ( ( complete_Sup_Sup @ B @ ( image2 @ C @ B @ F2 @ A4 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F2 @ A4 ) ) )
=> ( ( condit941137186595557371_above @ B @ ( image2 @ C @ B @ F2 @ A4 ) )
=> ( ( condit1013018076250108175_below @ B @ ( image2 @ C @ B @ F2 @ A4 ) )
=> ( ( member @ C @ A3 @ A4 )
=> ( ( F2 @ A3 )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_5883_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G2: B > A,B3: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ B3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( ord_less_eq @ A @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_5884_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_5885_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ A @ ( F2 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_5886_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ B3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_5887_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_5888_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( complete_Inf_Inf @ A @ S )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X3: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_5889_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_5890_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A4 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F2 @ ( A4 @ X3 ) )
@ I4 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_5891_bsqr__ofilter,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),D4: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) @ D4 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D4
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu: A] : ( field2 @ A @ R3 ) ) )
=> ( ~ ? [A8: A] :
( ( field2 @ A @ R3 )
= ( order_under @ A @ R3 @ A8 ) )
=> ? [A10: set @ A] :
( ( order_ofilter @ A @ R3 @ A10 )
& ( ord_less @ ( set @ A ) @ A10 @ ( field2 @ A @ R3 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D4
@ ( product_Sigma @ A @ A @ A10
@ ^ [Uu: A] : A10 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_5892_Refl__under__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( order_under @ A @ R3 @ A3 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_5893_ofilter__Restr__under,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( order_under @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ A3 )
= ( order_under @ A @ R3 @ A3 ) ) ) ) ) ).
% ofilter_Restr_under
thf(fact_5894_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
thf(fact_5895_well__ordering,axiom,
! [A: $tType] :
? [R6: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R6 ) @ R6 )
& ( ( field2 @ A @ R6 )
= ( top_top @ ( set @ A ) ) ) ) ).
% well_ordering
thf(fact_5896_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_5897_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 ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) ) ) ).
% underS_E
thf(fact_5898_underS__I,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( I != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ I @ ( order_underS @ A @ R @ J ) ) ) ) ).
% underS_I
thf(fact_5899_well__order__on__domain,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A4 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A4 )
& ( member @ A @ B2 @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_5900_underS__empty,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( order_underS @ A @ R3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_5901_Well__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_well_order_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% Well_order_Restr
thf(fact_5902_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_5903_underS__Field3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( field2 @ A @ R3 ) ) ) ).
% underS_Field3
thf(fact_5904_well__order__on__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R3 ) )
=> ( order_well_order_on @ A @ A4
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_5905_Field__Restr__ofilter,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
= A4 ) ) ) ).
% Field_Restr_ofilter
thf(fact_5906_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_5907_underS__incl__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_underS @ A @ R3 @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_5908_Linear__order__Well__order__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R3 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_5909_ofilter__Restr__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ A4 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_5910_ofilter__Restr__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% ofilter_Restr_Int
thf(fact_5911_bsqr__max2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A1: A,A22: A,B15: A,B24: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B15 @ B24 ) ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B15 @ B24 ) ) @ R3 ) ) ) ).
% bsqr_max2
thf(fact_5912_UNION__inj__on__ofilter,axiom,
! [C: $tType,A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ A ),F2: A > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( order_ofilter @ A @ R3 @ ( A4 @ I2 ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( inj_on @ A @ C @ F2 @ ( A4 @ I2 ) ) )
=> ( inj_on @ A @ C @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) ) ) ) ).
% UNION_inj_on_ofilter
thf(fact_5913_ofilter__subset__embedS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( order_ofilter @ A @ R3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_5914_ofilter__subset__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( order_ofilter @ A @ R3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_5915_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_5916_ordLess__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% ordLess_irreflexive
thf(fact_5917_ordLess__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We4044943003108391690rdLess @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R10 ) @ R10 )
& ? [X4: A > A2] : ( bNF_Wellorder_embedS @ A @ A2 @ R4 @ R10 @ X4 ) ) ) ) ) ).
% ordLess_def
thf(fact_5918_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R5 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_5919_underS__Restr__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ A3 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ A3 ) ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_5920_ofilter__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ ( field2 @ A @ R3 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_5921_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( order_ofilter @ A @ R3 @ B3 )
=> ( ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ ( id @ A ) ) )
& ( ( A4 = B3 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_5922_ordLess__iff__ordIso__Restr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X3 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ X3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_5923_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X3 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ X3 ) ) )
@ R5 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_5924_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% not_ordLess_ordIso
thf(fact_5925_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% not_ordLess_ordLeq
thf(fact_5926_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordLess_imp_ordLeq
thf(fact_5927_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_5928_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_5929_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_5930_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_5931_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
thf(fact_5932_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R3 @ R5 @ F2 )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R5 ) ) ) ).
% iso_forward
thf(fact_5933_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_5934_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_5935_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_5936_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R5 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R9 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_5937_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordIso_imp_ordLeq
thf(fact_5938_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% ordIso_iff_ordLeq
thf(fact_5939_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% ordIso_symmetric
thf(fact_5940_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R5: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R5 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_5941_ordIso__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordIso @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R10 ) @ R10 )
& ? [X4: A > A2] : ( bNF_Wellorder_iso @ A @ A2 @ R4 @ R10 @ X4 ) ) ) ) ) ).
% ordIso_def
thf(fact_5942_finite__well__order__on__ordIso,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( order_well_order_on @ A @ A4 @ R3 )
=> ( ( order_well_order_on @ A @ A4 @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% finite_well_order_on_ordIso
thf(fact_5943_ordLeq__total,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLeq_total
thf(fact_5944_ordLeq__reflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_reflexive
thf(fact_5945_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
& ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ).
% ordLeq_Well_order_simp
thf(fact_5946_exists__minim__Well__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
=> ( order_well_order_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
=> ? [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_5947_ordIso__reflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_reflexive
thf(fact_5948_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > B,A17: set @ B] :
( ( order_well_order_on @ A @ A4 @ R3 )
=> ( ( bij_betw @ A @ B @ F2 @ A4 @ A17 )
=> ? [R11: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A17 @ R11 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_5949_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ B @ B ),F: A > B] :
( ( bij_betw @ A @ B @ F @ ( field2 @ A @ R4 ) @ ( field2 @ B @ R10 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R4 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( field2 @ A @ R4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) @ R10 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_5950_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R5: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R5 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R5 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( 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 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_5951_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% not_ordLess_iff_ordLeq
thf(fact_5952_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% not_ordLeq_iff_ordLess
thf(fact_5953_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLess_or_ordLeq
thf(fact_5954_ofilter__subset__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( order_ofilter @ A @ R3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_5955_ofilter__embed,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R3 ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ R3
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_5956_ofilter__subset__embed,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A,B3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A4 )
=> ( ( order_ofilter @ A @ R3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_5957_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),F132: A > C,F232: B > C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R13 @ R24 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R24 @ R32 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R13 @ R32 @ F132 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R24 @ R32 @ F232 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F132 @ ( field2 @ A @ R13 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R24 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R32 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_5958_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ? [X_12: B > A] : ( bNF_Wellorder_embed @ B @ A @ R5 @ R3 @ X_12 ) ) ).
% ordLess_not_embed
thf(fact_5959_iso__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 ),F: A > A2] :
( ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R10 @ F )
& ( bij_betw @ A @ A2 @ F @ ( field2 @ A @ R4 ) @ ( field2 @ A2 @ R10 ) ) ) ) ) ).
% iso_defs(2)
thf(fact_5960_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R13 @ R24 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R13 @ R24 @ F2 )
=> ( ( image2 @ A @ B @ F2 @ ( field2 @ A @ R13 ) )
!= ( field2 @ B @ R24 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_5961_embedS__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_embedS @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 ),F: A > A2] :
( ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R10 @ F )
& ~ ( bij_betw @ A @ A2 @ F @ ( field2 @ A @ R4 ) @ ( field2 @ A2 @ R10 ) ) ) ) ) ).
% embedS_defs(2)
thf(fact_5962_ordLess__iff,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
& ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
& ~ ? [X4: B > A] : ( bNF_Wellorder_embed @ B @ A @ R5 @ R3 @ X4 ) ) ) ).
% ordLess_iff
thf(fact_5963_embed__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_embed @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 ),F: A > A2] :
! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R4 ) )
=> ( bij_betw @ A @ A2 @ F @ ( order_under @ A @ R4 @ X3 ) @ ( order_under @ A2 @ R10 @ ( F @ X3 ) ) ) ) ) ) ).
% embed_defs(2)
thf(fact_5964_ordLeq__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordLeq @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R10 ) @ R10 )
& ? [X4: A > A2] : ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R10 @ X4 ) ) ) ) ) ).
% ordLeq_def
thf(fact_5965_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),F2: B > A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( bNF_Wellorder_embed @ B @ A @ R5 @ R3 @ F2 )
=> ( bNF_Wellorder_iso @ B @ A @ R5
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F2 @ ( field2 @ B @ R5 ) )
@ ^ [Uu: A] : ( image2 @ B @ A @ F2 @ ( field2 @ B @ R5 ) ) ) )
@ F2 ) ) ) ) ).
% embed_implies_iso_Restr
thf(fact_5966_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X: C,X7: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
( ( member @ ( 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
@ ( complete_Sup_Sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
@ ^ [F: C > ( set @ ( set @ A ) )] : ( F @ X )
@ X7 ) ) ) )
@ Hbd )
@ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( 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 @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X7 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_5967_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( inj_on @ A @ B @ F2 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_We2720479622203943262_image @ A @ B @ R3 @ F2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image_ordIso
thf(fact_5968_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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_5969_card__of__Times1,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : A4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_5970_card__of__Times2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_5971_card__of__Pow,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( 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 @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ A4 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_5972_infinite__iff__card__of__nat,axiom,
! [A: $tType,A4: set @ A] :
( ( ~ ( finite_finite2 @ A @ A4 ) )
= ( member @ ( 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 @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_5973_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B3: set @ A,I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B3 )
=> ( ( member @ ( 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 @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( 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 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_5974_card__of__Times__same__infinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( member @ ( 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 @ A4
@ ^ [Uu: A] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_5975_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: 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 ) ) ) )
@ ( 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 @ A4
@ ^ [Uu: A] : B3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : A4 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_5976_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( 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 ) ) ) )
@ ( 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 @ C3
@ ^ [Uu: C] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C3
@ ^ [Uu: C] : B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_5977_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( 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 ) ) ) )
@ ( 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 @ A4
@ ^ [Uu: A] : C3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B3
@ ^ [Uu: B] : C3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_5978_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B3: A > ( set @ C )] :
( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B3 @ X2 ) ) ) @ ( 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 ) ) ) ) @ ( 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 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I4 @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_5979_card__of__Times3,axiom,
! [A: $tType,A4: set @ A] :
( member @ ( 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 @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_5980_card__of__refl,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( 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 @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_5981_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A4: 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 ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I4 @ A4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_5982_card__of__Pow__Func,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A4 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_5983_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
@ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A4
@ ( product_Sigma @ B @ C @ B3
@ ^ [Uu: B] : C3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A4 @ B3 )
@ ^ [Uu: A > B] : ( bNF_Wellorder_Func @ A @ C @ A4 @ C3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_5984_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( 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 ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_5985_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_5986_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_5987_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_5988_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B3
@ ^ [Uu: B] : A4 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_5989_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 )
@ C3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B3 @ C3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_5990_card__of__image,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B] : ( member @ ( 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 @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_5991_card__of__empty3,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_5992_card__of__empty,axiom,
! [B: $tType,A: $tType,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_5993_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ( finite_finite2 @ B @ B3 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_5994_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_5995_card__of__mono1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( member @ ( 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 @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_5996_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( 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
thf(fact_5997_card__of__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_5998_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A4 )
= ( finite_finite2 @ B @ B3 ) ) ) ).
% card_of_ordIso_finite
thf(fact_5999_card__of__mono2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( 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 @ R3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_6000_card__of__cong,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( 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 @ R3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R5 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_6001_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( member @ B @ ( F2 @ A8 ) @ B3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_6002_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),F: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu: product_prod @ A2 @ A2] :
? [A5: A,B4: A] :
( ( Uu
= ( product_Pair @ A2 @ A2 @ ( F @ A5 ) @ ( F @ B4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R4 ) ) ) ) ) ).
% dir_image_def
thf(fact_6003_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F3: B > A,B10: set @ B] : ( bij_betw @ B @ A @ F3 @ B10 @ A4 ) ) ).
% ex_bij_betw
thf(fact_6004_card__of__least,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A4 @ R3 )
=> ( member @ ( 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 @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_6005_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( 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 @ R3 ) ) @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_6006_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_6007_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( ? [F: A > B] : ( bij_betw @ A @ B @ F @ A4 @ B3 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_6008_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X: D] :
( ! [Y2: A] : ( member @ ( 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 @ ( S @ Y2 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( 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 @ ( comp @ A @ ( set @ B ) @ D @ S @ Rep @ X ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_6009_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G: B > A] :
( ( image2 @ B @ A @ G @ B3 )
= A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_6010_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B3: set @ A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( member @ ( 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 @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_6011_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( 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 @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_6012_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B3: set @ A,A4: set @ B] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F: B > A] :
( ( image2 @ B @ A @ F @ A4 )
= B3 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_6013_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A4: set @ A,C3: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ C3 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( 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 @ B5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_6014_card__of__Field__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 @ R3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_6015_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B3: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) @ B3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_6016_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),A4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( 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 ) ) ) )
@ ( 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 @ A4
@ ^ [Uu: C] : ( field2 @ A @ R3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A4
@ ^ [Uu: C] : ( field2 @ B @ R5 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_6017_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( 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 ) ) ) )
@ ( 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 @ R3 )
@ ^ [Uu: A] : C3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R5 )
@ ^ [Uu: B] : C3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_6018_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( ? [F: A > B] :
( ( inj_on @ A @ B @ F @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ B3 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_6019_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ ( field2 @ B @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( 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 @ B5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_6020_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( ~ ? [F: A > B] :
( ( inj_on @ A @ B @ F @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ B3 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_6021_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B3: set @ A,I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B3 )
=> ( ( member @ ( 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 @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( 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 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_6022_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [K7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K7 @ ( field2 @ A @ R4 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K7 @ R4 ) )
=> ( member @ ( 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 ) @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_6023_card__of__ordIso__subst,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ( member @ ( 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 @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_6024_SIGMA__CSUM,axiom,
! [B: $tType,A: $tType,I4: set @ A,As10: A > ( set @ B )] :
( ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I4 @ As10 ) )
= ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 )
@ ^ [I3: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( As10 @ I3 ) ) ) ) ).
% SIGMA_CSUM
thf(fact_6025_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B,B3: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( 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 @ B3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 @ A4
@ ^ [Uu: B] : B3 ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_6026_card__order__on__ordIso,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% card_order_on_ordIso
thf(fact_6027_Card__order__trans,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( Y != Z2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( ( X != Z2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_6028_Field__card__order,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ( field2 @ A @ R3 )
= ( top_top @ ( set @ A ) ) ) ) ).
% Field_card_order
thf(fact_6029_infinite__Card__order__limit,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
& ( A3 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R3 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_6030_ordLeq__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_refl
thf(fact_6031_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ).
% Card_order_ordIso
thf(fact_6032_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ).
% Card_order_ordIso2
thf(fact_6033_ordIso__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_refl
thf(fact_6034_card__of__unique,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_6035_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A6 @ R4 )
& ! [R10: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A6 @ R10 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R10 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_6036_card__order__dir__image,axiom,
! [B: $tType,A: $tType,F2: A > B,R3: set @ ( product_prod @ A @ A )] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( top_top @ ( set @ B ) ) @ ( bNF_We2720479622203943262_image @ A @ B @ R3 @ F2 ) ) ) ) ).
% card_order_dir_image
thf(fact_6037_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_6038_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_6039_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_6040_card__of__Field__ordIso,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 @ R3 ) ) @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_6041_dir__image,axiom,
! [B: $tType,A: $tType,F2: A > B,R3: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y2: A] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_We2720479622203943262_image @ A @ B @ R3 @ F2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image
thf(fact_6042_exists__minim__Card__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
=> ? [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_6043_Card__order__empty,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_6044_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
= ( finite_finite2 @ B @ A4 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_6045_card__of__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( member @ ( 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 @ R3 @ A3 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_6046_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B2: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( 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 @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_6047_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B,B3: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_6048_card__of__empty1,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_6049_Card__order__Pow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_6050_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_6051_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : ( field2 @ A @ R3 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_6052_Card__order__Times__same__infinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( member @ ( 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 @ R3 )
@ ^ [Uu: A] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_6053_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( 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 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_6054_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X3 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ X3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_6055_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As10: A > ( set @ B ),B3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R3 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R3 @ As10 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As10 @ ( field2 @ A @ R3 ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
& ( ord_less_eq @ ( set @ B ) @ B3 @ ( As10 @ X2 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_6056_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ B @ P4 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu: A] : ( field2 @ B @ P4 ) ) )
@ R3 )
@ ( 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 ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P4 )
@ ^ [Uu: B] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_6057_Csum__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Cardinal_Csum @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),Rs: A > ( set @ ( product_prod @ B @ B ) )] :
( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R4 )
@ ^ [I3: A] : ( field2 @ B @ ( Rs @ I3 ) ) ) ) ) ) ).
% Csum_def
thf(fact_6058_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( 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 @ A4 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_6059_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ? [X_1: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R3 @ X_1 ) ) ) ).
% ex_toCard_pred
thf(fact_6060_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As10: ( set @ A ) > ( set @ B ),B3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As10 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As10 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B3 @ ( As10 @ X2 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_6061_cardSuc__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_6062_cardSuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_6063_cardSuc__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( 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 @ R3 ) @ R5 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_6064_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( member @ ( 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 @ R3 ) @ R5 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_6065_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( 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 @ R3 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R5 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_6066_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( 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 @ R3 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R5 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_6067_cardSuc__least__aux,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R5 ) @ R5 )
=> ( ( member @ ( 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 ) ) ) @ R3 @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
=> ( member @ ( 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 @ R3 ) @ R5 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_6068_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( member @ ( 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 ) ) ) @ R5 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_6069_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R4: set @ ( product_prod @ A @ 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 ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ R10 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R14: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R14 ) @ R14 )
& ( member @ ( 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 ) ) ) @ R4 @ R14 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
=> ( member @ ( 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 ) ) ) @ R10 @ R14 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_6070_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R3 @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R3 ) ) ) ) ).
% toCard_pred_toCard
thf(fact_6071_toCard__inj,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B ),X: A,Y: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A4 @ R3 @ X )
= ( bNF_Greatest_toCard @ A @ B @ A4 @ R3 @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% toCard_inj
thf(fact_6072_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B ),B2: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ A @ B2 @ A4 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A4 @ R3 @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R3 @ B2 ) )
= B2 ) ) ) ) ).
% fromCard_toCard
thf(fact_6073_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As10: ( set @ A ) > ( set @ B ),B3: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As10 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As10 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B3 @ ( As10 @ X2 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_6074_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R13 @ R24 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca4139267488887388095finite @ A @ R13 )
=> ( bNF_Ca4139267488887388095finite @ B @ R24 ) ) ) ).
% cinfinite_mono
thf(fact_6075_Cinfinite__limit,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
& ( X != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X2 ) @ R3 ) ) ) ) ).
% Cinfinite_limit
thf(fact_6076_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R3: set @ ( product_prod @ A @ A ),X22: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ X22 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
& ( X1 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X2 ) @ R3 )
& ( X22 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X2 ) @ R3 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_6077_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R13 @ R24 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R13 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R13 ) @ R13 ) )
=> ( ( bNF_Ca4139267488887388095finite @ B @ R24 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R24 ) @ R24 ) ) ) ) ).
% Cinfinite_cong
thf(fact_6078_Cinfinite__limit__finite,axiom,
! [A: $tType,X7: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ X7 )
=> ( ( Xa2 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_6079_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A4: set @ A,R3: set @ ( product_prod @ B @ B ),B3: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( 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 @ B3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_6080_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set @ A,R3: set @ ( product_prod @ B @ B ),A4: A > ( set @ C )] :
( ( member @ ( 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A4 @ I4 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_6081_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ R3 ) @ ( 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 ) ) ) )
@ ( 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 )
@ ^ [I3: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ I3 ) ) )
@ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R3 ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% card_of_Csum_Times'
thf(fact_6082_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S3 ) @ S3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ S3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_6083_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 ) ) ) ) @ ( 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
thf(fact_6084_card__order__cprod,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R13 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( top_top @ ( set @ B ) ) @ R24 )
=> ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R13 @ R24 ) ) ) ) ).
% card_order_cprod
thf(fact_6085_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S3 ) @ S3 ) )
=> ( member @ ( 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 @ R3 @ S3 ) @ S3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_6086_cprod__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Cardinal_cprod @ A @ B )
= ( ^ [R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] :
( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R12 )
@ ^ [Uu: A] : ( field2 @ B @ R23 ) ) ) ) ) ).
% cprod_def
thf(fact_6087_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 @ P22 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_6088_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R13 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 ) @ ( bNF_Cardinal_cprod @ B @ C @ R13 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_mono1
thf(fact_6089_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_6090_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 @ P22 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_6091_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R13 ) @ ( 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 ) ) ) ) @ ( 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 @ R13 @ P22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_6092_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_6093_cprod__infinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ R3 @ R3 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% cprod_infinite
thf(fact_6094_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P4 ) @ P4 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ P4 @ Q4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_6095_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B3: set @ C] :
( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( member @ ( 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 @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B3 ) ) @ ( 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 ) ) ) )
@ ( 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 )
@ ^ [I3: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ I3 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_6096_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B ),P9: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) ) ) @ ( 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 @ P4 @ P9 ) @ ( bNF_Cardinal_cprod @ A @ A @ R3 @ R3 ) ) @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P4 @ P9 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_6097_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X2: B] : ( member @ ( 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 @ ( Fset @ X2 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X2: D] : ( member @ ( 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 @ ( Gset @ X2 ) ) @ 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 ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_6098_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P4 ) @ P4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( 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 @ R3 @ P4 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_6099_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R13 ) @ R13 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R24 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R24 ) @ R24 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R13 @ R24 ) )
& ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R13 @ R24 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R13 @ R24 ) ) ) ) ) ).
% Cinfinite_cprod2
thf(fact_6100_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_6101_czero__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( 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
thf(fact_6102_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca4139267488887388095finite @ B @ R3 )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% cinfinite_not_czero
thf(fact_6103_czeroE,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_6104_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_6105_czeroI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_6106_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_6107_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ Q4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 ) ) ) ) ) ).
% Cnotzero_mono
thf(fact_6108_Cinfinite__Cnotzero,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ) ).
% Cinfinite_Cnotzero
thf(fact_6109_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ ( member @ ( 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
thf(fact_6110_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R13 ) @ R13 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R24 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R24 ) @ R24 ) )
=> ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R13 @ R24 ) ) ) ) ).
% cinfinite_cprod2
thf(fact_6111_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 ) ) ) @ ( 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 ) ) ) ) @ ( 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
thf(fact_6112_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% cone_ordLeq_Cnotzero
thf(fact_6113_cexp__mono,axiom,
! [E: $tType,F4: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( member @ ( 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 @ F4 @ F4 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F4 @ F4 ) ) @ R24 @ ( bNF_Cardinal_czero @ F4 ) ) @ ( bNF_Wellorder_ordIso @ D @ F4 ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_6114_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ C @ C ),R32: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R13 ) @ R13 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R13 @ R24 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R13 @ ( bNF_Cardinal_cprod @ C @ B @ R24 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% cexp_cprod
thf(fact_6115_cexp__cone,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R3 @ bNF_Cardinal_cone ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% cexp_cone
thf(fact_6116_cone__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( 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
thf(fact_6117_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ B @ B ),S3: set @ ( product_prod @ C @ C ),T4: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R3 @ S3 ) @ T4 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R3 @ T4 ) @ ( bNF_Cardinal_cexp @ C @ A @ S3 @ T4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_6118_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R13 ) @ R13 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R24 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R24 ) @ R24 ) )
=> ( ( ~ ( member @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R32 @ R24 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R13 @ R24 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ B @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% cexp_cprod_ordLeq
thf(fact_6119_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P22 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R24 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_6120_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ Q4 ) @ ( bNF_Cardinal_cexp @ B @ C @ R13 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_mono1
thf(fact_6121_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( ( field2 @ A @ P22 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R24 )
= ( bot_bot @ ( set @ B ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_6122_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q4 @ ( bNF_Cardinal_cexp @ B @ A @ Q4 @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% ordLeq_cexp1
thf(fact_6123_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R24 ) @ R24 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_6124_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ Q4 ) @ ( bNF_Cardinal_cexp @ B @ C @ R13 @ Q4 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_cong1
thf(fact_6125_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_6126_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ~ ( member @ ( 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 @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_6127_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( member @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R24 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_6128_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( 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 @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R3 @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% ordLeq_cexp2
thf(fact_6129_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S3: set @ ( product_prod @ A @ A ),T4: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S3 ) @ S3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T4 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T4 ) @ T4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S3 @ T4 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T4 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_6130_ctwo__Cnotzero,axiom,
( ~ ( member @ ( 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
thf(fact_6131_ctwo__def,axiom,
( bNF_Cardinal_ctwo
= ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) ) ).
% ctwo_def
thf(fact_6132_ordLess__ctwo__cexp,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R3 @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% ordLess_ctwo_cexp
thf(fact_6133_ctwo__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( 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
thf(fact_6134_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% ctwo_ordLeq_Cinfinite
thf(fact_6135_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% ctwo_ordLess_Cinfinite
thf(fact_6136_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( 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 @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) ) ) ).
% cinfinite_cexp
thf(fact_6137_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( 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 @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) )
& ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) ) ) ) ).
% Cinfinite_cexp
thf(fact_6138_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B3: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( 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 ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_6139_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% natLeq_ordLeq_cinfinite
thf(fact_6140_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_6141_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R3 ) @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_6142_ctwo__ordLess__natLeq,axiom,
member @ ( 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
thf(fact_6143_card__of__Plus__Times__bool,axiom,
! [A: $tType,A4: 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 ) ) ) )
@ ( 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 @ A4 @ A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_6144_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: 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 ) ) ) ) @ ( 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 @ A4 @ B3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B3 @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_6145_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: 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 ) ) ) ) ) @ ( 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 @ A4 @ B3 ) @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A4 @ ( sum_Plus @ B @ C @ B3 @ C3 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_6146_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B3: set @ A,A4: set @ B] : ( member @ ( 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 @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_6147_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_6148_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: 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 ) ) ) ) )
@ ( 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 ) @ A4
@ ^ [Uu: A] : ( sum_Plus @ B @ C @ B3 @ C3 ) ) )
@ ( 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 @ A4
@ ^ [Uu: A] : B3 )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : C3 ) ) ) )
@ ( 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
thf(fact_6149_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( member @ ( 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 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_6150_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_6151_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_6152_natLeq__card__order,axiom,
bNF_Ca8970107618336181345der_on @ nat @ ( top_top @ ( set @ nat ) ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_card_order
thf(fact_6153_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_6154_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),A4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( 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 ) ) ) ) @ ( 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 @ A4 @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R5 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_6155_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( 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 ) ) ) ) @ ( 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 @ R3 ) @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R5 ) @ C3 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_6156_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),P4: set @ ( product_prod @ C @ C ),P9: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P4 @ P9 ) @ ( 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 ) ) ) ) @ ( 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 @ R3 ) @ ( field2 @ C @ P4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R5 ) @ ( field2 @ D @ P9 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_6157_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),A4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( 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 ) ) ) ) @ ( 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 @ A4 @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_6158_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( 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 ) ) ) ) @ ( 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 @ R3 ) @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R5 ) @ C3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_6159_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),P4: set @ ( product_prod @ C @ C ),P9: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P4 @ P9 ) @ ( 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 ) ) ) ) @ ( 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 @ R3 ) @ ( field2 @ C @ P4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R5 ) @ ( field2 @ D @ P9 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_6160_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C,D4: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( 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 @ C3 ) @ ( 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 ) ) ) ) @ ( 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 @ A4 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B3 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_6161_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( 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 ) ) ) ) @ ( 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 @ A4 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B3 @ C3 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_6162_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( 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 ) ) ) ) @ ( 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 @ C3 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C3 @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_6163_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C,D4: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( 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 @ C3 ) @ ( 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 ) ) ) ) @ ( 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 @ A4 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B3 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_6164_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_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 ) ) ) ) @ ( 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 @ A4 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B3 @ C3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_6165_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B3: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) @ ( bNF_Wellorder_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 ) ) ) ) @ ( 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 @ C3 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C3 @ B3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_6166_Field__natLeq,axiom,
( ( field2 @ nat @ bNF_Ca8665028551170535155natLeq )
= ( top_top @ ( set @ nat ) ) ) ).
% Field_natLeq
thf(fact_6167_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_6168_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( 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 @ A4 @ B3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( 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 ) ) ) @ ( 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 @ B3 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_6169_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( 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 @ A4 @ B3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_6170_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( 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 @ B3 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_6171_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C3: set @ A,A4: set @ B,B3: set @ C] :
( ~ ( finite_finite2 @ A @ C3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( 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 @ B3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C3 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( 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 @ A4 @ B3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C3 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_6172_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : ( member @ ( 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 @ A6 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_6173_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_6174_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) )
@ ^ [Uu: nat] :
( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_6175_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( 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 @ R3 ) @ ( field2 @ B @ P4 ) ) ) @ R3 ) @ ( 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 ) ) ) @ ( 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 @ P4 ) @ ( field2 @ A @ R3 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_6176_card__of__nat,axiom,
member @ ( 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
thf(fact_6177_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B15: B,B24: B,B3: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( ( B15 != B24 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B15 @ ( insert2 @ B @ B24 @ ( bot_bot @ ( set @ B ) ) ) ) @ B3 ) )
=> ( 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 ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_6178_card__of__Field__natLeq,axiom,
member @ ( 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
thf(fact_6179_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B,B3: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( 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 @ B3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( 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 @ A4 @ B3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_6180_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ B,B3: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( 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 @ B3 ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( 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 @ A4 @ B3 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_6181_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
!= ( member @ ( 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 @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_6182_UNIV__Plus__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% UNIV_Plus_UNIV
thf(fact_6183_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( ( sum_Plus @ A @ B @ A4 @ B3 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_6184_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B ),P9: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) ) ) @ ( 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 @ P4 @ P9 ) @ ( bNF_Cardinal_csum @ A @ A @ R3 @ R3 ) ) @ ( 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 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ C @ P4 @ P9 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% csum_dup
thf(fact_6185_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R13 ) @ R13 ) )
=> ( ~ ( 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 ) ) ) ) @ ( 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 @ R13 @ R24 ) @ ( 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 @ R13 @ R24 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R13 @ R24 ) ) ) ) ).
% csum_Cnotzero1
thf(fact_6186_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 ) ) ) ) ) @ ( 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
thf(fact_6187_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R13: set @ ( product_prod @ A @ A ),R24: 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 ) ) ) ) ) @ ( 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 @ R13 @ R24 ) @ ( bNF_Cardinal_csum @ C @ D @ R32 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R13 @ R32 ) @ ( bNF_Cardinal_csum @ B @ D @ R24 @ 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
thf(fact_6188_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R13: set @ ( product_prod @ A @ A ),R24: 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 ) ) ) ) ) @ ( 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 @ R13 @ R24 ) @ ( bNF_Cardinal_cprod @ A @ C @ R13 @ R32 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R13 @ ( bNF_Cardinal_csum @ B @ C @ R24 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% cprod_csum_distrib1
thf(fact_6189_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R13 @ R24 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R13 @ R24 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% cprod_csum_cexp
thf(fact_6190_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 ) ) ) ) @ ( 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
thf(fact_6191_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B ),T4: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( bNF_Cardinal_cfinite @ B @ S3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S3 ) @ S3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ C @ T4 )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T4 ) @ T4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R3 @ S3 ) @ T4 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R3 @ bNF_Cardinal_ctwo ) @ T4 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
thf(fact_6192_card__order__csum,axiom,
! [A: $tType,B: $tType,R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R13 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( top_top @ ( set @ B ) ) @ R24 )
=> ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R13 @ R24 ) ) ) ) ).
% card_order_csum
thf(fact_6193_Un__csum,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( member @ ( 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 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_6194_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 ) ) ) ) @ ( 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
thf(fact_6195_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 ) ) ) ) @ ( 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
thf(fact_6196_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_cong
thf(fact_6197_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R13 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 ) @ ( bNF_Cardinal_csum @ B @ C @ R13 @ Q4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_cong1
thf(fact_6198_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 @ P22 ) @ ( bNF_Cardinal_csum @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_cong2
thf(fact_6199_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R24: set @ ( product_prod @ D @ D )] :
( ( member @ ( 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 @ R13 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ R13 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_mono
thf(fact_6200_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R13 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 ) @ ( bNF_Cardinal_csum @ B @ C @ R13 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_mono1
thf(fact_6201_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( 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 @ R24 ) @ ( 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 ) ) ) ) @ ( 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 @ Q4 @ P22 ) @ ( bNF_Cardinal_csum @ C @ B @ Q4 @ R24 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_mono2
thf(fact_6202_Plus__csum,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: 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 ) ) ) ) @ ( 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 @ A4 @ B3 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% Plus_csum
thf(fact_6203_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),S3: set @ ( product_prod @ C @ C )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ T4 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T4 ) @ T4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R3 @ S3 ) @ T4 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R3 @ S3 ) @ T4 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% cprod_cexp_csum_cexp_Cinfinite
thf(fact_6204_csum__absorb1,axiom,
! [B: $tType,A: $tType,R24: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R24 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R24 ) @ R24 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R13 @ R24 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( 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 @ R24 @ R13 ) @ R24 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% csum_absorb1
thf(fact_6205_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R24: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R24 ) @ R24 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R13 @ R24 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R13 )
| ( bNF_Ca4139267488887388095finite @ A @ R24 ) )
=> ( member @ ( 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 @ R24 @ R13 ) @ R24 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% csum_absorb1'
thf(fact_6206_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R24: set @ ( product_prod @ A @ A ),R13: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R24 ) @ R24 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R13 @ R24 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R13 )
| ( bNF_Ca4139267488887388095finite @ A @ R24 ) )
=> ( member @ ( 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 @ R13 @ R24 ) @ R24 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% csum_absorb2'
thf(fact_6207_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P4 ) @ P4 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( 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 @ P4 @ Q4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% csum_cinfinite_bound
thf(fact_6208_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_6209_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J5: set @ A,I4: set @ B,F2: D > C,F5: B > ( filter @ D ),G5: A > ( filter @ C )] :
( ! [M3: A] :
( ( member @ A @ M3 @ J5 )
=> ? [X5: B] :
( ( member @ B @ X5 @ I4 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F2 @ ( F5 @ X5 ) ) @ ( G5 @ M3 ) ) ) )
=> ( filterlim @ D @ C @ F2 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G5 @ J5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F5 @ I4 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_6210_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X3: A] : X3 )
= ( ^ [F7: filter @ A] : F7 ) ) ).
% filtermap_id'
thf(fact_6211_filtermap__bot,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( filtermap @ B @ A @ F2 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_bot
thf(fact_6212_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,F5: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F2 @ F5 ) )
= ( eventually @ B
@ ^ [X3: B] : ( P @ ( F2 @ X3 ) )
@ F5 ) ) ).
% eventually_filtermap
thf(fact_6213_filtermap__sup,axiom,
! [A: $tType,B: $tType,F2: B > A,F13: filter @ B,F24: filter @ B] :
( ( filtermap @ B @ A @ F2 @ ( sup_sup @ ( filter @ B ) @ F13 @ F24 ) )
= ( sup_sup @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ F13 ) @ ( filtermap @ B @ A @ F2 @ F24 ) ) ) ).
% filtermap_sup
thf(fact_6214_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,F13: filter @ B,G2: C > A,F24: filter @ C] :
( ( filterlim @ A @ B @ F2 @ F13 @ ( filtermap @ C @ A @ G2 @ F24 ) )
= ( filterlim @ C @ B
@ ^ [X3: C] : ( F2 @ ( G2 @ X3 ) )
@ F13
@ F24 ) ) ).
% filterlim_filtermap
thf(fact_6215_filtermap__ident,axiom,
! [A: $tType,F5: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X3: A] : X3
@ F5 )
= F5 ) ).
% filtermap_ident
thf(fact_6216_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G2: C > B,F5: filter @ C] :
( ( filtermap @ B @ A @ F2 @ ( filtermap @ C @ B @ G2 @ F5 ) )
= ( filtermap @ C @ A
@ ^ [X3: C] : ( F2 @ ( G2 @ X3 ) )
@ F5 ) ) ).
% filtermap_filtermap
thf(fact_6217_filtermap__eq__strong,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ A,G5: filter @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( filtermap @ A @ B @ F2 @ F5 )
= ( filtermap @ A @ B @ F2 @ G5 ) )
= ( F5 = G5 ) ) ) ).
% filtermap_eq_strong
thf(fact_6218_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,F5: filter @ B] :
( ( ( filtermap @ B @ A @ F2 @ F5 )
= ( bot_bot @ ( filter @ A ) ) )
= ( F5
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_6219_map__filter__on__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ B ) ) ).
% map_filter_on_UNIV
thf(fact_6220_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F2: nat > A] :
( ( filtermap @ nat @ A @ F2 @ ( at_top @ nat ) )
!= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_sequentually_ne_bot
thf(fact_6221_filtermap__inf,axiom,
! [A: $tType,B: $tType,F2: B > A,F13: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ ( inf_inf @ ( filter @ B ) @ F13 @ F24 ) ) @ ( inf_inf @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ F13 ) @ ( filtermap @ B @ A @ F2 @ F24 ) ) ) ).
% filtermap_inf
thf(fact_6222_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G2: A > B,F5: filter @ B,G5: filter @ A,F2: B > A] :
( ( filterlim @ A @ B @ G2 @ F5 @ G5 )
=> ( ( filterlim @ B @ A @ F2 @ G5 @ F5 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ ( G2 @ X3 ) )
= X3 )
@ G5 )
=> ( ( filtermap @ B @ A @ F2 @ F5 )
= G5 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_6223_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,F5: C > ( filter @ B ),B3: set @ C] :
( ( filtermap @ B @ A @ F2 @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B3 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B4: C] : ( filtermap @ B @ A @ F2 @ ( F5 @ B4 ) )
@ B3 ) ) ) ).
% filtermap_SUP
thf(fact_6224_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P2: B > $o] :
( eventually @ A
@ ^ [X3: A] : ( P2 @ ( F @ X3 ) )
@ F7 ) ) ) ) ).
% filtermap_def
thf(fact_6225_filtermap__mono__strong,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ A,G5: filter @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F2 @ F5 ) @ ( filtermap @ A @ B @ F2 @ G5 ) )
= ( ord_less_eq @ ( filter @ A ) @ F5 @ G5 ) ) ) ).
% filtermap_mono_strong
thf(fact_6226_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N )
= one2 ) ) ).
% num_of_nat_One
thf(fact_6227_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,F5: C > ( filter @ B ),B3: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B3 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B4: C] : ( filtermap @ B @ A @ F2 @ ( F5 @ B4 ) )
@ B3 ) ) ) ).
% filtermap_INF
thf(fact_6228_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F5: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F5 @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A5: A] : ( product_Pair @ A @ B @ A5 @ X )
@ F5 ) ) ).
% prod_filter_principal_singleton2
thf(fact_6229_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_6230_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F5: filter @ A,G5: filter @ B,H9: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F5 @ G5 ) @ H9 )
= ( filtermap @ ( 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 )
@ ^ [X3: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y3: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G5 @ H9 ) ) ) ) ).
% prod_filter_assoc
thf(fact_6231_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A4: filter @ A,B3: filter @ B] :
( ( ( prod_filter @ A @ B @ A4 @ B3 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( filter @ A ) ) )
| ( B3
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_6232_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F5: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F5 @ F5 ) )
= ( ? [Q: A > $o] :
( ( eventually @ A @ Q @ F5 )
& ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( P @ ( product_Pair @ A @ A @ X3 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_6233_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F5: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F5 @ G5 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F5 )
& ( eventually @ B @ Pg @ G5 )
& ! [X3: A,Y3: B] :
( ( Pf @ X3 )
=> ( ( Pg @ Y3 )
=> ( P @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_6234_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A4: filter @ A,B3: filter @ B,C3: filter @ A,D4: filter @ B] :
( ( A4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B3
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A4 @ B3 ) @ ( prod_filter @ A @ B @ C3 @ D4 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( filter @ B ) @ B3 @ D4 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_6235_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G5: filter @ B,F5: filter @ A,G2: A > C,H9: filter @ C] :
( ( filterlim @ A @ B @ F2 @ G5 @ F5 )
=> ( ( filterlim @ A @ C @ G2 @ H9 @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X3: A] : ( product_Pair @ B @ C @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( prod_filter @ B @ C @ G5 @ H9 )
@ F5 ) ) ) ).
% filterlim_Pair
thf(fact_6236_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,G2: C > B,F5: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ F5 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F2 @ F5 ) @ ( filtermap @ C @ B @ G2 @ F5 ) ) ) ).
% filtermap_Pair
thf(fact_6237_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A4: set @ A,B3: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A4 ) @ ( principal @ B @ B3 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B3 ) ) ) ).
% principal_prod_principal
thf(fact_6238_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F7: filter @ A,G8: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P2: A > $o,Q: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( P2 @ X3 )
& ( Q @ Y3 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P2: A > $o,Q: B > $o] :
( ( eventually @ A @ P2 @ F7 )
& ( eventually @ B @ Q @ G8 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_6239_eventually__prod__sequentially,axiom,
! [P: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N11: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N11 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ N11 @ N2 )
=> ( P @ ( product_Pair @ nat @ nat @ N2 @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_6240_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: filter @ A,Q2: B > $o,G5: filter @ B] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ B @ Q2 @ G5 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X3 ) )
& ( Q2 @ ( product_snd @ A @ B @ X3 ) ) )
@ ( prod_filter @ A @ B @ F5 @ G5 ) ) ) ) ).
% eventually_prodI
thf(fact_6241_eventually__prod1,axiom,
! [A: $tType,B: $tType,B3: filter @ A,P: B > $o,A4: filter @ B] :
( ( B3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X3: B,Y3: A] : ( P @ X3 ) )
@ ( prod_filter @ B @ A @ A4 @ B3 ) )
= ( eventually @ B @ P @ A4 ) ) ) ).
% eventually_prod1
thf(fact_6242_eventually__prod2,axiom,
! [A: $tType,B: $tType,A4: filter @ A,P: B > $o,B3: filter @ B] :
( ( A4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A] : P )
@ ( prod_filter @ A @ B @ A4 @ B3 ) )
= ( eventually @ B @ P @ B3 ) ) ) ).
% eventually_prod2
thf(fact_6243_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J5: set @ A,A4: filter @ B,B3: A > ( filter @ C )] :
( ( J5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A4 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B3 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ A4 @ ( B3 @ I3 ) )
@ J5 ) ) ) ) ).
% prod_filter_INF2
thf(fact_6244_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I4: set @ A,A4: A > ( filter @ B ),B3: filter @ C] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A4 @ I4 ) ) @ B3 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ ( A4 @ I3 ) @ B3 )
@ I4 ) ) ) ) ).
% prod_filter_INF1
thf(fact_6245_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I4: set @ A,J5: set @ B,A4: A > ( filter @ C ),B3: B > ( filter @ D )] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A4 @ I4 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B3 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I3: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A4 @ I3 ) @ ( B3 @ J3 ) )
@ J5 ) )
@ I4 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_6246_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_6247_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F5: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F5 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F5 ) ) ).
% prod_filter_principal_singleton
thf(fact_6248_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_6249_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S8: set @ ( product_prod @ B @ A ),K4: B] :
( eps_Opt @ A
@ ^ [V2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K4 @ V2 ) @ S8 ) ) ) ) ).
% set_to_map_def
thf(fact_6250_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [A5: A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_6251_some__opt__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ^ [Y3: A] : Y3 = X )
= ( some @ A @ X ) ) ).
% some_opt_eq_trivial
thf(fact_6252_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_6253_alt__ex1E_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X5 ) ) )
=> ~ ( ? [X_1: A] : ( P @ X_1 )
=> ~ ( uniq @ A @ P ) ) ) ).
% alt_ex1E'
thf(fact_6254_ex1__iff__ex__Uniq,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P2: A > $o] :
( ? [X4: A] : ( P2 @ X4 )
& ( uniq @ A @ P2 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_6255_inj__on__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F: A > B,A6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( uniq @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( ( F @ X3 )
= ( F @ Y3 ) ) ) ) ) ) ) ).
% inj_on_iff_Uniq
thf(fact_6256_pairwise__disjnt__iff,axiom,
! [A: $tType,A19: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A19 )
= ( ! [X3: A] :
( uniq @ ( set @ A )
@ ^ [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A19 )
& ( member @ A @ X3 @ X4 ) ) ) ) ) ).
% pairwise_disjnt_iff
thf(fact_6257_the1__equality_H,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( uniq @ A @ P )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality'
thf(fact_6258_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs2 )
& ( ( set2 @ A @ Xs2 )
= A4 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_6259_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
thf(fact_6260_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F2
@ ^ [X3: A] : ( member @ A @ X3 @ S )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_6261_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ A4 )
= ( F2 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_6262_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( insert2 @ A @ A3 @ A4 ) )
= ( F2 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_6263_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( bot_bot @ ( set @ A ) ) )
= Top ) ) ).
% bounded_quasi_semilattice_set.empty
thf(fact_6264_bounded__quasi__semilattice__set_Ounion,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A4: set @ A,B3: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( F2 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ A4 ) @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ B3 ) ) ) ) ).
% bounded_quasi_semilattice_set.union
thf(fact_6265_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ B ),F: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: B,A23: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F @ A12 ) @ ( F @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R2 ) ) ) ) ) ).
% relImage_def
thf(fact_6266_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_6267_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F2: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F2 @ A4 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F2 @ A4 ) ) )
=> ( ( bNF_Gr4221423524335903396lImage @ B @ A @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A4 @ R @ F2 ) @ F2 )
= R ) ) ).
% relImage_relInvImage
thf(fact_6268_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A4: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_6269_mset__set_Ofolding__on__axioms,axiom,
! [A: $tType] : ( finite_folding_on @ A @ ( multiset @ A ) @ ( top_top @ ( set @ A ) ) @ ( add_mset @ A ) ) ).
% mset_set.folding_on_axioms
thf(fact_6270_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: A > B > B,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F2 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% folding_on.empty
thf(fact_6271_card_Ofolding__on__axioms,axiom,
! [A: $tType] :
( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : suc ) ).
% card.folding_on_axioms
thf(fact_6272_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite_folding_on @ A @ ( list @ A ) @ ( top_top @ ( set @ A ) )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_6273_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A6: set @ A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: A,A23: A] :
( ( Uu
= ( product_Pair @ A @ A @ A12 @ A23 ) )
& ( member @ A @ A12 @ A6 )
& ( member @ A @ A23 @ A6 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A12 ) @ ( F @ A23 ) ) @ R2 ) ) ) ) ) ).
% relInvImage_def
thf(fact_6274_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F2: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Gr7122648621184425601vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R @ F2 ) @ F2 ) ) ).
% relInvImage_UNIV_relImage
thf(fact_6275_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A4: set @ A,X: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A4 )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_6276_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),B3: set @ A,A4: set @ B,F2: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B3
@ ^ [Uu: A] : B3 ) )
=> ( ( bNF_Gr7122648621184425601vImage @ B @ A @ A4 @ R @ F2 )
= ( relcomp @ B @ A @ B @ ( bNF_Gr @ B @ A @ A4 @ F2 ) @ ( relcomp @ A @ A @ B @ R @ ( converse @ B @ A @ ( bNF_Gr @ B @ A @ A4 @ F2 ) ) ) ) ) ) ).
% relInvImage_Gr
thf(fact_6277_relImage__Gr,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ( bNF_Gr4221423524335903396lImage @ A @ B @ R @ F2 )
= ( relcomp @ B @ A @ B @ ( converse @ A @ B @ ( bNF_Gr @ A @ B @ A4 @ F2 ) ) @ ( relcomp @ A @ A @ B @ R @ ( bNF_Gr @ A @ B @ A4 @ F2 ) ) ) ) ) ).
% relImage_Gr
thf(fact_6278_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R3 ) ) ).
% converse_iff
thf(fact_6279_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
thf(fact_6280_converse__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_UNIV
thf(fact_6281_pair__set__inverse,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [B4: A,A5: B] : ( P @ A5 @ B4 ) ) ) ) ).
% pair_set_inverse
thf(fact_6282_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_6283_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_6284_in__listrel1__converse,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R3 ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% in_listrel1_converse
thf(fact_6285_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R3 ) ) ) ).
% converseI
thf(fact_6286_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R3 ) )
=> ~ ! [X2: B,Y2: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y2 @ X2 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ R3 ) ) ) ).
% converseE
thf(fact_6287_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R3 ) ) ).
% converseD
thf(fact_6288_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R3 ) )
= ( ? [A5: A,B4: B] :
( ( A1 = B4 )
& ( A22 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_6289_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R3 ) ) ).
% converse.cases
thf(fact_6290_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_6291_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_6292_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,X3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).
% converse_unfold
thf(fact_6293_converse__Int,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S3 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R3 ) @ ( converse @ B @ A @ S3 ) ) ) ).
% converse_Int
thf(fact_6294_converse__Times,axiom,
! [B: $tType,A: $tType,A4: set @ B,B3: set @ A] :
( ( converse @ B @ A
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B3 ) )
= ( product_Sigma @ A @ B @ B3
@ ^ [Uu: A] : A4 ) ) ).
% converse_Times
thf(fact_6295_converse__Un,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R3 ) @ ( converse @ B @ A @ S3 ) ) ) ).
% converse_Un
thf(fact_6296_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R3: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X3: C] : ( converse @ B @ A @ ( R3 @ X3 ) )
@ S ) ) ) ).
% converse_UNION
thf(fact_6297_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R3: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X3: C] : ( converse @ B @ A @ ( R3 @ X3 ) )
@ S ) ) ) ).
% converse_INTER
thf(fact_6298_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A4: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R3 @ A4 ) @ B3 )
= ( ord_less_eq @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_6299_irrefl__tranclI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R3 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% irrefl_tranclI
thf(fact_6300_trans__wf__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( wf @ A @ R3 )
= ( ! [A5: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R3 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu: A] : ( image @ A @ A @ ( converse @ A @ A @ R3 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_6301_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),A4: set @ C,B3: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R3 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B3 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( image @ B @ A @ R3 @ ( B3 @ X3 ) )
@ A4 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_6302_trans__reflclI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( trans @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ).
% trans_reflclI
thf(fact_6303_trans__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( trans @ A @ S3 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) ) ) ) ).
% trans_Int
thf(fact_6304_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ).
% single_valued_inter1
thf(fact_6305_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S @ R ) ) ) ).
% single_valued_inter2
thf(fact_6306_lexord__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R3 ) )
=> ( ( trans @ A @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_trans
thf(fact_6307_transD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 ) ) ) ) ).
% transD
thf(fact_6308_transE,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 ) ) ) ) ).
% transE
thf(fact_6309_transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R3 ) ) )
=> ( trans @ A @ R3 ) ) ).
% transI
thf(fact_6310_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X3: A,Y3: A,Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z5 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z5 ) @ R4 ) ) ) ) ) ).
% trans_def
thf(fact_6311_single__valuedD,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ R3 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_6312_single__valuedI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y2: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R3 )
=> ( Y2 = Z3 ) ) )
=> ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedI
thf(fact_6313_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 )
=> ! [Z5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R4 )
=> ( Y3 = Z5 ) ) ) ) ) ).
% single_valued_def
thf(fact_6314_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_6315_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_6316_lenlex__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R3 ) )
=> ( ( trans @ A @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% lenlex_trans
thf(fact_6317_trans__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( trans @ A @ R3 )
=> ( trans @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) ) ) ).
% trans_Restr
thf(fact_6318_single__valued__confluent,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( single_valued @ A @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_6319_under__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R3 @ A3 ) @ ( order_under @ A @ R3 @ B2 ) ) ) ) ).
% under_incr
thf(fact_6320_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_6321_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ A4 )
=> ( ( transitive_rtrancl @ A @ A4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) ) ) ).
% trans_rtrancl_eq_reflcl
thf(fact_6322_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Y13: A] :
! [Z5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z5 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y13 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa3 ) @ R4 ) )
@ Z5 ) )
@ X3 ) ) ) ) ).
% trans_join
thf(fact_6323_underS__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_underS @ A @ R3 @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_6324_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B3: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B3 ) ) ) ) ).
% Image_Int_eq
thf(fact_6325_wf__finite__segments,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R3 )
=> ( ( trans @ A @ R3 )
=> ( ! [X2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R3 ) ) )
=> ( wf @ A @ R3 ) ) ) ) ).
% wf_finite_segments
thf(fact_6326_ord__to__filter__compat,axiom,
! [A: $tType,R0: set @ ( product_prod @ A @ A )] :
( bNF_Wellorder_compat @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ 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 ) ) @ ( insert2 @ ( set @ ( product_prod @ A @ A ) ) @ R0 @ ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) ) )
@ ^ [Uu: 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 ) ) @ ( insert2 @ ( set @ ( product_prod @ A @ A ) ) @ R0 @ ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) )
@ ( bNF_We413866401316099525erIncl @ A @ R0 )
@ ( bNF_We8469521843155493636filter @ A @ R0 ) ) ).
% ord_to_filter_compat
thf(fact_6327_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,F5: filter @ A,G5: filter @ B] :
( ( rel_filter @ A @ B @ R @ F5 @ G5 )
=> ~ ! [Z11: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) @ Z11 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_fst @ A @ B ) @ Z11 )
= F5 )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_snd @ A @ B ) @ Z11 )
!= G5 ) ) ) ) ).
% rel_filter.cases
thf(fact_6328_bot__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( rel_filter @ A @ B @ A4 @ ( bot_bot @ ( filter @ A ) ) @ ( bot_bot @ ( filter @ B ) ) ) ).
% bot_filter_parametric
thf(fact_6329_sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A4 ) @ ( rel_filter @ A @ B @ A4 ) ) @ ( sup_sup @ ( filter @ A ) ) @ ( sup_sup @ ( filter @ B ) ) ) ).
% sup_filter_parametric
thf(fact_6330_compat__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_compat @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ A2 @ A2 ),F: A > A2] :
! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R4 )
=> ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F @ A5 ) @ ( F @ B4 ) ) @ R10 ) ) ) ) ).
% compat_def
thf(fact_6331_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Z6: filter @ ( product_prod @ A @ B ),F5: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) @ Z6 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_fst @ A @ B ) @ Z6 )
= F5 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_snd @ A @ B ) @ Z6 )
= G5 )
=> ( rel_filter @ A @ B @ R @ F5 @ G5 ) ) ) ) ).
% rel_filter.intros
thf(fact_6332_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R2: A > B > $o,F7: filter @ A,G8: filter @ B] :
? [Z10: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) @ Z10 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) @ ( product_fst @ A @ B ) @ Z10 )
= F7 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) @ ( product_snd @ A @ B ) @ Z10 )
= G8 ) ) ) ) ).
% rel_filter.simps
thf(fact_6333_ord__to__filter__def,axiom,
! [A: $tType] :
( ( bNF_We8469521843155493636filter @ A )
= ( ^ [R02: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] : ( image2 @ A @ A @ ( fChoice @ ( A > A ) @ ( bNF_Wellorder_embed @ A @ A @ R4 @ R02 ) ) @ ( field2 @ A @ R4 ) ) ) ) ).
% ord_to_filter_def
thf(fact_6334_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F2 @ S )
= ( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ S )
& ( ( F2 @ Y3 )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F2 @ S ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_6335_some__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some_equality
thf(fact_6336_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y3: A] : Y3 = X )
= X ) ).
% some_eq_trivial
thf(fact_6337_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_6338_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_6339_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert2 @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_6340_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A4: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A4 )
=> ( ( ? [X4: A] : ( P @ X4 ) )
= A4 ) ) ).
% verit_sko_ex'
thf(fact_6341_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P5: A > $o] :
! [X6: A] : ( P5 @ X6 ) )
= ( ^ [P2: A > $o] :
( P2
@ ( fChoice @ A
@ ^ [X3: A] :
~ ( P2 @ X3 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_6342_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A4: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
= A4 )
=> ( ( ! [X4: A] : ( P @ X4 ) )
= A4 ) ) ).
% verit_sko_forall'
thf(fact_6343_verit__sko__forall_H_H,axiom,
! [A: $tType,B3: A,A4: A,P: A > $o] :
( ( B3 = A4 )
=> ( ( ( fChoice @ A @ P )
= A4 )
= ( ( fChoice @ A @ P )
= B3 ) ) ) ).
% verit_sko_forall''
thf(fact_6344_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X4: A] : ( P @ X4 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_6345_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P7: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X2: A] :
( ( P @ X2 )
= ( P7 @ X2 ) )
=> ( ( ? [X4: A] : ( P7 @ X4 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_6346_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
=> ( ( ! [X4: A] : ( P @ X4 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_6347_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P7: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
= ( P7 @ X2 ) )
=> ( ( ! [X4: A] : ( P7 @ X4 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_6348_someI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q2: A > $o] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( Q2 @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_6349_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_6350_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( Q2 @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_6351_someI2__bex,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( P @ X5 ) )
=> ( ! [X2: A] :
( ( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) )
=> ( Q2 @ X2 ) )
=> ( Q2
@ ( fChoice @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_6352_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X4: A] : ( P @ X4 ) ) ) ).
% some_eq_ex
thf(fact_6353_some1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( ( P @ A3 )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some1_equality
thf(fact_6354_some__in__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
@ A4 )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_6355_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
@ S ) ) ).
% some_elem
thf(fact_6356_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P2: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B4 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_6357_card__of__def,axiom,
! [A: $tType] :
( ( bNF_Ca6860139660246222851ard_of @ A )
= ( ^ [A6: set @ A] : ( fChoice @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca8970107618336181345der_on @ A @ A6 ) ) ) ) ).
% card_of_def
thf(fact_6358_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A15: A,X_12: B] : ( P @ A15 @ X_12 )
=> ( ! [B17: B,B25: B] :
( ? [A8: A] : ( P @ A8 @ B17 )
=> ( ? [A8: A] : ( P @ A8 @ B25 )
=> ( B17 = B25 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A5: A] :
? [X4: B] : ( P @ A5 @ X4 ) )
@ ( the @ B
@ ^ [B4: B] :
? [A5: A] : ( P @ A5 @ B4 ) ) ) ) ) ).
% some_theI
thf(fact_6359_equiv__Eps__preserves,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ X7 ) )
@ A4 ) ) ) ).
% equiv_Eps_preserves
thf(fact_6360_equiv__Eps__in,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ X7 ) )
@ X7 ) ) ) ).
% equiv_Eps_in
thf(fact_6361_cardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca8387033319878233205ardSuc @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( fChoice @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6246979054910435723ardSuc @ A @ R4 ) ) ) ) ).
% cardSuc_def
thf(fact_6362_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F: B > A,P2: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F @ P2 ) ) ) ) ) ).
% arg_min_def
thf(fact_6363_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_6364_toCard__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_toCard @ A @ B )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ B @ B )] : ( fChoice @ ( A > B ) @ ( bNF_Gr1419584066657907630d_pred @ A @ B @ A6 @ R4 ) ) ) ) ).
% toCard_def
thf(fact_6365_proj__Eps,axiom,
! [A: $tType,A4: set @ A,R3: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A4 @ R3 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A4 @ R3 ) )
=> ( ( equiv_proj @ A @ A @ R3
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ X7 ) ) )
= X7 ) ) ) ).
% proj_Eps
thf(fact_6366_fromCard__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr5436034075474128252omCard @ A @ B )
= ( ^ [A6: set @ A,R4: set @ ( product_prod @ B @ B ),K4: B] :
( fChoice @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ A6 )
& ( ( bNF_Greatest_toCard @ A @ B @ A6 @ R4 @ B4 )
= K4 ) ) ) ) ) ).
% fromCard_def
thf(fact_6367_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P2: A > $o] :
( if @ ( option @ A )
@ ? [X4: A] : ( P2 @ X4 )
@ ( some @ A @ ( fChoice @ A @ P2 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_6368_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ A ),X3: B] :
( fChoice @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R2 ) ) ) ) ).
% fun_of_rel_def
thf(fact_6369_to__nat__def,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( to_nat @ A )
= ( fChoice @ ( A > nat )
@ ^ [F: A > nat] : ( inj_on @ A @ nat @ F @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% to_nat_def
thf(fact_6370_inj__to__nat,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( inj_on @ A @ nat @ ( to_nat @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_to_nat
thf(fact_6371_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C3: set @ A] :
( ( pred_chain @ A @ A4 @ P @ C3 )
=> ( ~ ( pred_maxchain @ A @ A4 @ P @ C3 )
=> ( ( pred_chain @ A @ A4 @ P
@ ( fChoice @ ( set @ A )
@ ^ [D6: set @ A] :
( ( pred_chain @ A @ A4 @ P @ D6 )
& ( ord_less @ ( set @ A ) @ C3 @ D6 ) ) ) )
& ( ord_less @ ( set @ A ) @ C3
@ ( fChoice @ ( set @ A )
@ ^ [D6: set @ A] :
( ( pred_chain @ A @ A4 @ P @ D6 )
& ( ord_less @ ( set @ A ) @ C3 @ D6 ) ) ) ) ) ) ) ).
% pred_on.not_maxchain_Some
thf(fact_6372_map__comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( map_comp @ B @ C @ A )
= ( ^ [F: B > ( option @ C ),G: A > ( option @ B ),K4: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F @ ( G @ K4 ) ) ) ) ).
% map_comp_def
thf(fact_6373_map__comp__empty_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,M: C > ( option @ B )] :
( ( map_comp @ C @ B @ A @ M
@ ^ [X3: A] : ( none @ C ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% map_comp_empty(1)
thf(fact_6374_map__comp__empty_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,M: C > ( option @ B )] :
( ( map_comp @ B @ D @ C
@ ^ [X3: B] : ( none @ D )
@ M )
= ( ^ [X3: C] : ( none @ D ) ) ) ).
% map_comp_empty(2)
thf(fact_6375_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) )
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ D6 ) ) ) )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ D6 ) ) ) ) ) ) ) ).
% subset.not_maxchain_Some
thf(fact_6376_pred__on_Osuc__def,axiom,
! [A: $tType] :
( ( pred_suc @ A )
= ( ^ [A6: set @ A,P2: A > A > $o,C7: set @ A] :
( if @ ( set @ A )
@ ( ~ ( pred_chain @ A @ A6 @ P2 @ C7 )
| ( pred_maxchain @ A @ A6 @ P2 @ C7 ) )
@ C7
@ ( fChoice @ ( set @ A )
@ ^ [D6: set @ A] :
( ( pred_chain @ A @ A6 @ P2 @ D6 )
& ( ord_less @ ( set @ A ) @ C7 @ D6 ) ) ) ) ) ) ).
% pred_on.suc_def
thf(fact_6377_univ__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Greatest_univ @ B @ A )
= ( ^ [F: B > A,X4: set @ B] :
( F
@ ( fChoice @ B
@ ^ [X3: B] : ( member @ B @ X3 @ X4 ) ) ) ) ) ).
% univ_def
thf(fact_6378_subset_Osuc__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( ( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
| ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
= C3 ) )
& ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
| ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ D6 ) ) ) ) ) ) ).
% subset.suc_def
thf(fact_6379_flip__pred,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ A @ B ),R: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) )
@ A4 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% flip_pred
thf(fact_6380_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_6381_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X3: A,Y3: A] : X3 != Y3 )
= ( ^ [X3: A,Y3: A] : X3 != Y3 ) ) ).
% conversep_noteq
thf(fact_6382_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: B,Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ ( converse @ A @ B @ R3 ) ) ) ) ).
% conversep_converse_eq
thf(fact_6383_converse__meet,axiom,
! [A: $tType,B: $tType,R3: B > A > $o,S3: B > A > $o] :
( ( conversep @ B @ A @ ( inf_inf @ ( B > A > $o ) @ R3 @ S3 ) )
= ( inf_inf @ ( A > B > $o ) @ ( conversep @ B @ A @ R3 ) @ ( conversep @ B @ A @ S3 ) ) ) ).
% converse_meet
thf(fact_6384_converse__join,axiom,
! [A: $tType,B: $tType,R3: B > A > $o,S3: B > A > $o] :
( ( conversep @ B @ A @ ( sup_sup @ ( B > A > $o ) @ R3 @ S3 ) )
= ( sup_sup @ ( A > B > $o ) @ ( conversep @ B @ A @ R3 ) @ ( conversep @ B @ A @ S3 ) ) ) ).
% converse_join
thf(fact_6385_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ).
% converse_def
thf(fact_6386_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P6: product_prod @ C @ D] : ( insert2 @ D @ ( product_snd @ C @ D @ P6 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_6387_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_6388_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_6389_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( insert2 @ A @ ( product_fst @ A @ B @ P6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_6390_prod_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R15: A > C > $o,R25: B > D > $o,A5: product_prod @ A @ B,B4: product_prod @ C @ D] :
? [Z5: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z5
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X3: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R15 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z5 )
= A5 )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z5 )
= B4 ) ) ) ) ).
% prod.in_rel
thf(fact_6391_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_6392_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: A > B > $o,R22: C > D > $o,A3: A,B2: C,C2: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
= ( ( R1 @ A3 @ C2 )
& ( R22 @ B2 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_6393_prod_Orel__map_I2_J,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,S1a: A > E > $o,S2a: B > F4 > $o,X: product_prod @ A @ B,G1: C > E,G22: D > F4,Y: product_prod @ C @ D] :
( ( basic_rel_prod @ A @ E @ B @ F4 @ S1a @ S2a @ X @ ( product_map_prod @ C @ E @ D @ F4 @ G1 @ G22 @ Y ) )
= ( basic_rel_prod @ A @ C @ B @ D
@ ^ [X3: A,Y3: C] : ( S1a @ X3 @ ( G1 @ Y3 ) )
@ ^ [X3: B,Y3: D] : ( S2a @ X3 @ ( G22 @ Y3 ) )
@ X
@ Y ) ) ).
% prod.rel_map(2)
thf(fact_6394_prod_Orel__map_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,S1b: E > C > $o,S2b: F4 > D > $o,I1: A > E,I22: B > F4,X: product_prod @ A @ B,Y: product_prod @ C @ D] :
( ( basic_rel_prod @ E @ C @ F4 @ D @ S1b @ S2b @ ( product_map_prod @ A @ E @ B @ F4 @ I1 @ I22 @ X ) @ Y )
= ( basic_rel_prod @ A @ C @ B @ D
@ ^ [X3: A] : ( S1b @ ( I1 @ X3 ) )
@ ^ [X3: B] : ( S2b @ ( I22 @ X3 ) )
@ X
@ Y ) ) ).
% prod.rel_map(1)
thf(fact_6395_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: A > B > $o,R22: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ A1 @ A22 )
=> ~ ! [A8: A,B7: B,C4: C] :
( ( A1
= ( product_Pair @ A @ C @ A8 @ C4 ) )
=> ! [D2: D] :
( ( A22
= ( product_Pair @ B @ D @ B7 @ D2 ) )
=> ( ( R1 @ A8 @ B7 )
=> ~ ( R22 @ C4 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_6396_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R15: A > B > $o,R25: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
? [A5: A,B4: B,C5: C,D5: D] :
( ( A12
= ( product_Pair @ A @ C @ A5 @ C5 ) )
& ( A23
= ( product_Pair @ B @ D @ B4 @ D5 ) )
& ( R15 @ A5 @ B4 )
& ( R25 @ C5 @ D5 ) ) ) ) ).
% rel_prod.simps
thf(fact_6397_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > B > $o,A3: A,B2: B,R22: C > D > $o,C2: C,D3: D] :
( ( R1 @ A3 @ B2 )
=> ( ( R22 @ C2 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( product_Pair @ B @ D @ B2 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_6398_rel__prod__conv,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R15: A > C > $o,R25: B > D > $o] :
( product_case_prod @ A @ B @ ( ( product_prod @ C @ D ) > $o )
@ ^ [A5: A,B4: B] :
( product_case_prod @ C @ D @ $o
@ ^ [C5: C,D5: D] :
( ( R15 @ A5 @ C5 )
& ( R25 @ B4 @ D5 ) ) ) ) ) ) ).
% rel_prod_conv
thf(fact_6399_int_Oid__abs__transfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int
@ ( basic_rel_prod @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ pcr_int
@ ^ [X3: product_prod @ nat @ nat] : X3
@ abs_Integ ) ).
% int.id_abs_transfer
thf(fact_6400_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : ( single_valuedp @ A @ B @ ( bot_bot @ ( A > B > $o ) ) ) ).
% single_valuedp_bot
thf(fact_6401_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R4: A > B > $o] :
! [X3: A] : ( uniq @ B @ ( R4 @ X3 ) ) ) ) ).
% single_valuedp_iff_Uniq
thf(fact_6402_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B3: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A4 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B3 @ ( basic_rel_prod @ A @ B @ C @ D @ A4 @ B3 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_6403_prod_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R15: A > C > $o,R25: B > D > $o] :
( relcompp @ ( product_prod @ A @ B ) @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( conversep @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X3: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R15 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X3: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R15 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% prod.rel_compp_Grp
thf(fact_6404_revg_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [A8: A,As4: list @ A] :
( ( X
= ( cons @ A @ A8 @ As4 ) )
=> ( ( Y
= ( revg @ A @ As4 @ ( cons @ A @ A8 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A8 @ As4 ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_6405_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > C > $o,S: C > B > $o,T2: C > B > $o] :
( ( relcompp @ A @ C @ B @ R @ ( sup_sup @ ( C > B > $o ) @ S @ T2 ) )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ R @ S ) @ ( relcompp @ A @ C @ B @ R @ T2 ) ) ) ).
% relcompp_distrib
thf(fact_6406_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: A > C > $o,T2: A > C > $o,R: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( sup_sup @ ( A > C > $o ) @ S @ T2 ) @ R )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ S @ R ) @ ( relcompp @ A @ C @ B @ T2 @ R ) ) ) ).
% relcompp_distrib2
thf(fact_6407_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R: A > C > $o] :
( ( relcompp @ A @ C @ B @ R @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_6408_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_6409_OO__Grp__alt,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ C,F2: C > A,G2: C > B] :
( ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ ( bNF_Grp @ C @ A @ A4 @ F2 ) ) @ ( bNF_Grp @ C @ B @ A4 @ G2 ) )
= ( ^ [X3: A,Y3: B] :
? [Z5: C] :
( ( member @ C @ Z5 @ A4 )
& ( ( F2 @ Z5 )
= X3 )
& ( ( G2 @ Z5 )
= Y3 ) ) ) ) ).
% OO_Grp_alt
thf(fact_6410_Grp__UNIV__id,axiom,
! [A: $tType,F2: A > A] :
( ( F2
= ( id @ A ) )
=> ( ( relcompp @ A @ A @ A @ ( conversep @ A @ A @ ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) )
= ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ).
% Grp_UNIV_id
thf(fact_6411_list_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( list_all2 @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F2 ) )
= ( bNF_Grp @ ( list @ A ) @ ( list @ B )
@ ( collect @ ( list @ A )
@ ^ [X3: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ X3 ) @ A4 ) )
@ ( map @ A @ B @ F2 ) ) ) ).
% list.rel_Grp
thf(fact_6412_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
@ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S3 ) )
= ( ^ [X3: A,Y3: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ ( relcomp @ A @ B @ C @ R3 @ S3 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_6413_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A6: set @ A,F: A > B,A5: A,B4: B] :
( ( B4
= ( F @ A5 ) )
& ( member @ A @ A5 @ A6 ) ) ) ) ).
% Grp_def
thf(fact_6414_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R2: A > C > $o,S8: C > B > $o,X3: A,Z5: B] :
? [Y3: C] :
( ( R2 @ X3 @ Y3 )
& ( S8 @ Y3 @ Z5 ) ) ) ) ).
% OO_def
thf(fact_6415_Grp__UNIV__idI,axiom,
! [A: $tType,X: A,Y: A] :
( ( X = Y )
=> ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) @ X @ Y ) ) ).
% Grp_UNIV_idI
thf(fact_6416_eq__alt,axiom,
! [A: $tType] :
( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) ) ) ).
% eq_alt
thf(fact_6417_rel__filter__Grp,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( rel_filter @ A @ B @ ( bNF_Grp @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
= ( bNF_Grp @ ( filter @ A ) @ ( filter @ B ) @ ( top_top @ ( set @ ( filter @ A ) ) ) @ ( filtermap @ A @ B @ F2 ) ) ) ).
% rel_filter_Grp
thf(fact_6418_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_Grp @ A @ B @ A4 @ F2 ) )
= ( bNF_Grp @ ( D > A ) @ ( D > B )
@ ( collect @ ( D > A )
@ ^ [X3: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) @ A4 ) )
@ ( comp @ A @ B @ D @ F2 ) ) ) ).
% fun.rel_Grp
thf(fact_6419_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: A > C > $o,R3: D > C > B > $o,I4: set @ D] :
( ( relcompp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( C > B > $o ) @ ( image2 @ D @ ( C > B > $o ) @ R3 @ I4 ) ) )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I3: D] : ( relcompp @ A @ C @ B @ S3 @ ( R3 @ I3 ) )
@ I4 ) ) ) ).
% relcompp_SUP_distrib
thf(fact_6420_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R3: D > A > C > $o,I4: set @ D,S3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R3 @ I4 ) ) @ S3 )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I3: D] : ( relcompp @ A @ C @ B @ ( R3 @ I3 ) @ S3 )
@ I4 ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_6421_prod_Orel__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A18: set @ A,F1: A > C,A25: set @ B,F22: B > D] :
( ( basic_rel_prod @ A @ C @ B @ D @ ( bNF_Grp @ A @ C @ A18 @ F1 ) @ ( bNF_Grp @ B @ D @ A25 @ F22 ) )
= ( bNF_Grp @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B] :
( ( ord_less_eq @ ( set @ A ) @ ( basic_fsts @ A @ B @ X3 ) @ A18 )
& ( ord_less_eq @ ( set @ B ) @ ( basic_snds @ A @ B @ X3 ) @ A25 ) ) )
@ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 ) ) ) ).
% prod.rel_Grp
thf(fact_6422_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 )
@ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S2 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_6423_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: A > B,Abs: B > A,G2: D > C,P: C > $o,H3: C > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ D @ C @ B @ A @ $o @ G2 @ Abs @ ( bNF_Grp @ C @ A @ ( collect @ C @ P ) @ H3 ) )
= ( bNF_Grp @ D @ B
@ ( collect @ D
@ ^ [X3: D] : ( P @ ( G2 @ X3 ) ) )
@ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H3 ) @ G2 ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_6424_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: A > B,Abs: B > A,F2: C > D,P: D > $o,H3: D > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ D @ A @ B @ $o @ F2 @ Rep @ ( bNF_Grp @ D @ B @ ( collect @ D @ P ) @ H3 ) )
= ( bNF_Grp @ C @ A
@ ( collect @ C
@ ^ [X3: C] : ( P @ ( F2 @ X3 ) ) )
@ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H3 ) @ F2 ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_6425_vimage2p__def,axiom,
! [B: $tType,E: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ D @ B @ E @ C )
= ( ^ [F: A > D,G: B > E,R2: D > E > C,X3: A,Y3: B] : ( R2 @ ( F @ X3 ) @ ( G @ Y3 ) ) ) ) ).
% vimage2p_def
thf(fact_6426_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R2: A > B > $o] :
( relcompp @ ( list @ A ) @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( conversep @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_6427_Abs__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: A > B,Abs1: B > A,Rep22: C > D,Abs22: D > C,R: B > D > $o] :
( ( type_definition @ A @ B @ Rep1 @ Abs1 @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep22 @ Abs22 @ ( top_top @ ( set @ D ) ) )
=> ( bNF_rel_fun @ B @ D @ A @ C @ R @ ( bNF_vimage2p @ A @ B @ C @ D @ $o @ Rep1 @ Rep22 @ R ) @ Abs1 @ Abs22 ) ) ) ).
% Abs_transfer
thf(fact_6428_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R )
= ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_6429_vimage2p__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ C @ B @ D @ $o )
= ( ^ [F: A > C,G: B > D,P2: C > D > $o] : ( relcompp @ A @ C @ B @ ( bNF_Grp @ A @ C @ ( top_top @ ( set @ A ) ) @ F ) @ ( relcompp @ C @ D @ B @ P2 @ ( conversep @ B @ D @ ( bNF_Grp @ B @ D @ ( top_top @ ( set @ B ) ) @ G ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_6430_vimage2p__relcompp__converse,axiom,
! [E: $tType,C: $tType,D: $tType,A: $tType,F4: $tType,B: $tType,Rep: A > B,Abs: B > A,F2: C > E,G2: D > F4,R: B > E > $o,S: B > F4 > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ E @ D @ F4 @ $o @ F2 @ G2 @ ( relcompp @ E @ B @ F4 @ ( conversep @ B @ E @ R ) @ S ) )
= ( relcompp @ C @ A @ D @ ( conversep @ A @ C @ ( bNF_vimage2p @ A @ B @ C @ E @ $o @ Rep @ F2 @ R ) ) @ ( bNF_vimage2p @ A @ B @ D @ F4 @ $o @ Rep @ G2 @ S ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_6431_Quotient__alt__def5,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R2: A > A > $o,Abs2: A > B,Rep2: B > A,T7: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ T7 @ ( bNF_Grp @ A @ B @ ( top_top @ ( set @ A ) ) @ Abs2 ) )
& ( ord_less_eq @ ( B > A > $o ) @ ( bNF_Grp @ B @ A @ ( top_top @ ( set @ B ) ) @ Rep2 ) @ ( conversep @ A @ B @ T7 ) )
& ( R2
= ( relcompp @ A @ B @ A @ T7 @ ( conversep @ A @ B @ T7 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_6432_acyclic__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( ( transitive_acyclic @ A @ R3 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% acyclic_insert
thf(fact_6433_acyclic__empty,axiom,
! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% acyclic_empty
thf(fact_6434_acyclicP__converse,axiom,
! [A: $tType,R3: A > A > $o] :
( ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( conversep @ A @ A @ R3 ) ) ) )
= ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ).
% acyclicP_converse
thf(fact_6435_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,T2: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( T2
= ( ^ [X3: B,Y3: A] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( quotient @ B @ A
@ ^ [Y5: B,Z4: B] : Y5 = Z4
@ Abs
@ Rep
@ T2 ) ) ) ).
% UNIV_typedef_to_Quotient
thf(fact_6436_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R3: set @ ( product_prod @ B @ B ),F2: B > A] :
( ! [A8: B,B7: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A8 @ B7 ) @ R3 )
=> ( ord_less @ A @ ( F2 @ B7 ) @ ( F2 @ A8 ) ) )
=> ( transitive_acyclic @ B @ R3 ) ) ) ).
% acyclicI_order
thf(fact_6437_Quotient__cr__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( T2
= ( ^ [X3: A,Y3: B] :
( ( R @ X3 @ X3 )
& ( ( Abs @ X3 )
= Y3 ) ) ) ) ) ).
% Quotient_cr_rel
thf(fact_6438_Quotient__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R2: A > A > $o,Abs2: A > B,Rep2: B > A,T7: A > B > $o] :
( ! [A5: B] :
( ( Abs2 @ ( Rep2 @ A5 ) )
= A5 )
& ! [A5: B] : ( R2 @ ( Rep2 @ A5 ) @ ( Rep2 @ A5 ) )
& ! [R4: A,S2: A] :
( ( R2 @ R4 @ S2 )
= ( ( R2 @ R4 @ R4 )
& ( R2 @ S2 @ S2 )
& ( ( Abs2 @ R4 )
= ( Abs2 @ S2 ) ) ) )
& ( T7
= ( ^ [X3: A,Y3: B] :
( ( R2 @ X3 @ X3 )
& ( ( Abs2 @ X3 )
= Y3 ) ) ) ) ) ) ) ).
% Quotient_def
thf(fact_6439_QuotientI,axiom,
! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R: B > B > $o,T2: B > A > $o] :
( ! [A8: A] :
( ( Abs @ ( Rep @ A8 ) )
= A8 )
=> ( ! [A8: A] : ( R @ ( Rep @ A8 ) @ ( Rep @ A8 ) )
=> ( ! [R6: B,S9: B] :
( ( R @ R6 @ S9 )
= ( ( R @ R6 @ R6 )
& ( R @ S9 @ S9 )
& ( ( Abs @ R6 )
= ( Abs @ S9 ) ) ) )
=> ( ( T2
= ( ^ [X3: B,Y3: A] :
( ( R @ X3 @ X3 )
& ( ( Abs @ X3 )
= Y3 ) ) ) )
=> ( quotient @ B @ A @ R @ Abs @ Rep @ T2 ) ) ) ) ) ).
% QuotientI
thf(fact_6440_acyclic__union_I2_J,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B3 ) )
=> ( transitive_acyclic @ A @ B3 ) ) ).
% acyclic_union(2)
thf(fact_6441_acyclic__union_I1_J,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B3 ) )
=> ( transitive_acyclic @ A @ A4 ) ) ).
% acyclic_union(1)
thf(fact_6442_cyclicE,axiom,
! [A: $tType,G2: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ G2 )
=> ~ ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ G2 ) ) ) ).
% cyclicE
thf(fact_6443_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_6444_acyclicI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( transitive_acyclic @ A @ R3 ) ) ).
% acyclicI
thf(fact_6445_acyclic__insert__cyclic,axiom,
! [A: $tType,G2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( transitive_acyclic @ A @ G2 )
=> ( ~ ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ G2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ G2 ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_6446_Quotient__eq__onp__type__copy,axiom,
! [B: $tType,A: $tType,Abs: A > B,Rep: B > A,Cr: A > B > $o] :
( ( quotient @ A @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Abs
@ Rep
@ Cr )
=> ( type_definition @ B @ A @ Rep @ Abs @ ( top_top @ ( set @ A ) ) ) ) ).
% Quotient_eq_onp_type_copy
thf(fact_6447_Quotient__crel__typecopy,axiom,
! [B: $tType,A: $tType,Abs: A > B,Rep: B > A,T2: A > B > $o] :
( ( quotient @ A @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Abs
@ Rep
@ T2 )
=> ( T2
= ( ^ [X3: A,Y3: B] :
( X3
= ( Rep @ Y3 ) ) ) ) ) ).
% Quotient_crel_typecopy
thf(fact_6448_su__rel__fun_Of__def,axiom,
! [A: $tType,B: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A4: A] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( F2 @ A4 )
= ( the @ B
@ ^ [B5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B5 ) @ F5 ) ) ) ) ).
% su_rel_fun.f_def
thf(fact_6449_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B] :
( ! [A10: A,B10: B,B8: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A10 @ B10 ) @ F5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A10 @ B8 ) @ F5 )
=> ( B10 = B8 ) ) )
=> ( ! [A10: A,P3: $o] :
( ! [B18: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A10 @ B18 ) @ F5 )
=> P3 )
=> P3 )
=> ( ! [A10: A] :
( ( F2 @ A10 )
= ( the @ B
@ ^ [B5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A10 @ B5 ) @ F5 ) ) )
=> ( su_rel_fun @ A @ B @ F5 @ F2 ) ) ) ) ).
% su_rel_fun.intro
thf(fact_6450_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A4: A] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ~ ! [B10: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B10 ) @ F5 ) ) ).
% su_rel_fun.surjective
thf(fact_6451_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A4: A,B3: B,B16: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ F5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B16 ) @ F5 )
=> ( B3 = B16 ) ) ) ) ).
% su_rel_fun.unique
thf(fact_6452_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A4: A,B3: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ F5 )
=> ( B3
= ( F2 @ A4 ) ) ) ) ).
% su_rel_fun.repr2
thf(fact_6453_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A4: A] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ ( F2 @ A4 ) ) @ F5 ) ) ).
% su_rel_fun.repr1
thf(fact_6454_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A4: A,B3: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( ( F2 @ A4 )
= B3 )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ F5 ) ) ) ).
% su_rel_fun.repr
thf(fact_6455_su__rel__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( su_rel_fun @ A @ B )
= ( ^ [F7: set @ ( product_prod @ A @ B ),F: A > B] :
( ! [A6: A,B5: B,B19: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ F7 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B19 ) @ F7 )
=> ( B5 = B19 ) ) )
& ! [A6: A,P2: $o] :
( ! [B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ F7 )
=> P2 )
=> P2 )
& ! [A6: A] :
( ( F @ A6 )
= ( the @ B
@ ^ [B5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ F7 ) ) ) ) ) ) ).
% su_rel_fun_def
thf(fact_6456_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_6457_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R4 ) ) ) ) ) ).
% aboveS_def
thf(fact_6458_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
thf(fact_6459_mult__cancel,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X7: multiset @ A,Z6: multiset @ A,Y4: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X7 @ Z6 ) @ ( plus_plus @ ( multiset @ A ) @ Y4 @ Z6 ) ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X7 @ Y4 ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel
thf(fact_6460_mult__cancel__max,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X7: multiset @ A,Y4: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X7 @ Y4 ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X7 @ Y4 ) @ ( minus_minus @ ( multiset @ A ) @ Y4 @ X7 ) ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_max
thf(fact_6461_mult__cancel__add__mset,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),Uu3: A,X7: multiset @ A,Y4: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu3 @ X7 ) @ ( add_mset @ A @ Uu3 @ Y4 ) ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X7 @ Y4 ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_6462_multp__code__iff__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > A > $o,N4: multiset @ A,M4: multiset @ A] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X2: A,Y2: A] :
( ( P @ X2 @ Y2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R ) )
=> ( ( multp_code @ A @ P @ N4 @ M4 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N4 @ M4 ) @ ( mult @ A @ R ) ) ) ) ) ) ).
% multp_code_iff_mult
thf(fact_6463_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > A > $o,N4: multiset @ A,M4: multiset @ A] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X2: A,Y2: A] :
( ( P @ X2 @ Y2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R ) )
=> ( ( multeqp_code @ A @ P @ N4 @ M4 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N4 @ M4 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
thf(fact_6464_mult__implies__one__step,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),M4: multiset @ A,N4: multiset @ A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M4 @ N4 ) @ ( mult @ A @ R3 ) )
=> ? [I9: multiset @ A,J6: multiset @ A] :
( ( N4
= ( plus_plus @ ( multiset @ A ) @ I9 @ J6 ) )
& ? [K8: multiset @ A] :
( ( M4
= ( plus_plus @ ( multiset @ A ) @ I9 @ K8 ) )
& ( J6
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_mset @ A @ K8 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_6465_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A3: A,M4: multiset @ A] :
( ~ ( member @ A @ A3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M4 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_6466_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_6467_set__mset__eq__empty__iff,axiom,
! [A: $tType,M4: multiset @ A] :
( ( ( set_mset @ A @ M4 )
= ( bot_bot @ ( set @ A ) ) )
= ( M4
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% set_mset_eq_empty_iff
thf(fact_6468_set__mset__union,axiom,
! [A: $tType,M4: multiset @ A,N4: multiset @ A] :
( ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M4 @ N4 ) )
= ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ M4 ) @ ( set_mset @ A @ N4 ) ) ) ).
% set_mset_union
thf(fact_6469_in__Inf__multiset__iff,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: A] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( member @ A @ X @ ( set_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) ) )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A4 )
=> ( member @ A @ X @ ( set_mset @ A @ X3 ) ) ) ) ) ) ).
% in_Inf_multiset_iff
thf(fact_6470_image__mset__cong,axiom,
! [B: $tType,A: $tType,M4: multiset @ A,F2: A > B,G2: A > B] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ M4 ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_mset @ A @ B @ F2 @ M4 )
= ( image_mset @ A @ B @ G2 @ M4 ) ) ) ).
% image_mset_cong
thf(fact_6471_in__image__mset,axiom,
! [A: $tType,B: $tType,Y: A,F2: B > A,M4: multiset @ B] :
( ( member @ A @ Y @ ( set_mset @ A @ ( image_mset @ B @ A @ F2 @ M4 ) ) )
= ( member @ A @ Y @ ( image2 @ B @ A @ F2 @ ( set_mset @ B @ M4 ) ) ) ) ).
% in_image_mset
thf(fact_6472_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M4: multiset @ ( product_prod @ A @ B ),F2: A > B > C,G2: A > B > C] :
( ! [X2: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M4 ) )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ M4 )
= ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G2 ) @ M4 ) ) ) ).
% image_mset_cong_pair
thf(fact_6473_infinite__set__mset__mset__set,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( set_mset @ A @ ( mset_set @ A @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_set_mset_mset_set
thf(fact_6474_set__mset__Inf,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( set_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ A4 ) ) ) ) ).
% set_mset_Inf
thf(fact_6475_set__mset__single,axiom,
! [A: $tType,B2: A] :
( ( set_mset @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_mset_single
thf(fact_6476_one__step__implies__mult,axiom,
! [A: $tType,J5: multiset @ A,K5: multiset @ A,R3: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
( ( J5
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ K5 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R3 ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K5 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J5 ) ) @ ( mult @ A @ R3 ) ) ) ) ).
% one_step_implies_mult
thf(fact_6477_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N11: multiset @ A,M9: multiset @ A] :
? [A5: A,M0: multiset @ A,K7: multiset @ A] :
( ( M9
= ( add_mset @ A @ A5 @ M0 ) )
& ( N11
= ( plus_plus @ ( multiset @ A ) @ M0 @ K7 ) )
& ! [B4: A] :
( ( member @ A @ B4 @ ( set_mset @ A @ K7 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R4 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_6478_less__add,axiom,
! [A: $tType,N4: multiset @ A,A3: A,M02: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N4 @ ( add_mset @ A @ A3 @ M02 ) ) @ ( mult1 @ A @ R3 ) )
=> ( ? [M10: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M10 @ M02 ) @ ( mult1 @ A @ R3 ) )
& ( N4
= ( add_mset @ A @ A3 @ M10 ) ) )
| ? [K8: multiset @ A] :
( ! [B13: A] :
( ( member @ A @ B13 @ ( set_mset @ A @ K8 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B13 @ A3 ) @ R3 ) )
& ( N4
= ( plus_plus @ ( multiset @ A ) @ M02 @ K8 ) ) ) ) ) ).
% less_add
thf(fact_6479_not__less__empty,axiom,
! [A: $tType,M4: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R3 ) ) ).
% not_less_empty
thf(fact_6480_mult1__union,axiom,
! [A: $tType,B3: multiset @ A,D4: multiset @ A,R3: set @ ( product_prod @ A @ A ),C3: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B3 @ D4 ) @ ( mult1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ B3 ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ D4 ) ) @ ( mult1 @ A @ R3 ) ) ) ).
% mult1_union
thf(fact_6481_mult1E,axiom,
! [A: $tType,N4: multiset @ A,M4: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N4 @ M4 ) @ ( mult1 @ A @ R3 ) )
=> ~ ! [A8: A,M03: multiset @ A] :
( ( M4
= ( add_mset @ A @ A8 @ M03 ) )
=> ! [K8: multiset @ A] :
( ( N4
= ( plus_plus @ ( multiset @ A ) @ M03 @ K8 ) )
=> ~ ! [B13: A] :
( ( member @ A @ B13 @ ( set_mset @ A @ K8 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B13 @ A8 ) @ R3 ) ) ) ) ) ).
% mult1E
thf(fact_6482_mult1I,axiom,
! [A: $tType,M4: multiset @ A,A3: A,M02: multiset @ A,N4: multiset @ A,K5: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( M4
= ( add_mset @ A @ A3 @ M02 ) )
=> ( ( N4
= ( plus_plus @ ( multiset @ A ) @ M02 @ K5 ) )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ ( set_mset @ A @ K5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N4 @ M4 ) @ ( mult1 @ A @ R3 ) ) ) ) ) ).
% mult1I
thf(fact_6483_mult1__lessE,axiom,
! [A: $tType,N4: multiset @ A,M4: multiset @ A,R3: A > A > $o] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N4 @ M4 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) )
=> ( ( asymp @ A @ R3 )
=> ~ ! [A8: A,M03: multiset @ A] :
( ( M4
= ( add_mset @ A @ A8 @ M03 ) )
=> ! [K8: multiset @ A] :
( ( N4
= ( plus_plus @ ( multiset @ A ) @ M03 @ K8 ) )
=> ( ~ ( member @ A @ A8 @ ( set_mset @ A @ K8 ) )
=> ~ ! [B13: A] :
( ( member @ A @ B13 @ ( set_mset @ A @ K8 ) )
=> ( R3 @ B13 @ A8 ) ) ) ) ) ) ) ).
% mult1_lessE
thf(fact_6484_sum__wcount__Int,axiom,
! [A: $tType,A4: set @ A,F2: A > nat,N4: multiset @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( wcount @ A @ F2 @ N4 ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( set_mset @ A @ N4 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ ( wcount @ A @ F2 @ N4 ) @ A4 ) ) ) ).
% sum_wcount_Int
thf(fact_6485_asymp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% asymp_greater
thf(fact_6486_multiset_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R2: A > B > $o,A5: multiset @ A,B4: multiset @ B] :
? [Z5: multiset @ ( product_prod @ A @ B )] :
( ( member @ ( multiset @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X3: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) ) )
& ( ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A5 )
& ( ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B4 ) ) ) ) ).
% multiset.in_rel
thf(fact_6487_multiset_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R2: A > B > $o] :
( relcompp @ ( multiset @ A ) @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ B )
@ ( conversep @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ A )
@ ( bNF_Grp @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ A )
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X3: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ B )
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X3: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% multiset.rel_compp_Grp
thf(fact_6488_multiset_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: multiset @ A,G2: B > C,Y: multiset @ B] :
( ( rel_mset @ A @ C @ Sa @ X @ ( image_mset @ B @ C @ G2 @ Y ) )
= ( rel_mset @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G2 @ Y3 ) )
@ X
@ Y ) ) ).
% multiset.rel_map(2)
thf(fact_6489_multiset_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: multiset @ A,Y: multiset @ B] :
( ( rel_mset @ C @ B @ Sb @ ( image_mset @ A @ C @ I @ X ) @ Y )
= ( rel_mset @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% multiset.rel_map(1)
thf(fact_6490_multiset_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( rel_mset @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F2 ) )
= ( bNF_Grp @ ( multiset @ A ) @ ( multiset @ B )
@ ( collect @ ( multiset @ A )
@ ^ [X3: multiset @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_mset @ A @ X3 ) @ A4 ) )
@ ( image_mset @ A @ B @ F2 ) ) ) ).
% multiset.rel_Grp
thf(fact_6491_smsI,axiom,
! [A4: multiset @ ( product_prod @ nat @ nat ),B3: multiset @ ( product_prod @ nat @ nat ),Z6: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B3 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z6 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z6 @ B3 ) ) @ ms_strict ) ) ).
% smsI
thf(fact_6492_wmsI,axiom,
! [A4: multiset @ ( product_prod @ nat @ nat ),B3: multiset @ ( product_prod @ nat @ nat ),Z6: multiset @ ( product_prod @ nat @ nat )] :
( ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B3 ) ) @ fun_max_strict )
| ( ( A4
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( B3
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z6 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z6 @ B3 ) ) @ ms_weak ) ) ).
% wmsI
thf(fact_6493_ms__weak__def,axiom,
( ms_weak
= ( sup_sup @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ms_strict @ ( id2 @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% ms_weak_def
thf(fact_6494_ms__reduction__pair,axiom,
fun_reduction_pair @ ( 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
thf(fact_6495_ms__strictI,axiom,
! [Z6: multiset @ ( product_prod @ nat @ nat ),Z14: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B3: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z6 @ Z14 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B3 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z6 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z14 @ B3 ) ) @ ms_strict ) ) ) ).
% ms_strictI
thf(fact_6496_ms__weakI1,axiom,
! [Z6: multiset @ ( product_prod @ nat @ nat ),Z14: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B3: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z6 @ Z14 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B3 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z6 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z14 @ B3 ) ) @ ms_weak ) ) ) ).
% ms_weakI1
thf(fact_6497_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 ) ) @ ( 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
thf(fact_6498_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 ) ) ) ) )
=> ~ ! [X2: product_prod @ nat @ nat,Y2: product_prod @ nat @ nat,X8: multiset @ ( product_prod @ nat @ nat )] :
( ( A1
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X8 ) )
=> ! [Y11: multiset @ ( product_prod @ nat @ nat )] :
( ( A22
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y11 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y2 ) @ fun_pair_leq )
=> ~ ( pw_leq @ X8 @ Y11 ) ) ) ) ) ) ).
% pw_leq.cases
thf(fact_6499_pw__leq_Osimps,axiom,
( pw_leq
= ( ^ [A12: multiset @ ( product_prod @ nat @ nat ),A23: multiset @ ( product_prod @ nat @ nat )] :
( ( ( A12
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A23
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
| ? [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat,X4: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
( ( A12
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X4 ) )
& ( A23
= ( 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 ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
& ( pw_leq @ X4 @ Y9 ) ) ) ) ) ).
% pw_leq.simps
thf(fact_6500_pw__leq__step,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X7: multiset @ ( product_prod @ nat @ nat ),Y4: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( 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 @ X7 @ Y4 )
=> ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X7 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y4 ) ) ) ) ).
% pw_leq_step
thf(fact_6501_ms__weakI2,axiom,
! [Z6: multiset @ ( product_prod @ nat @ nat ),Z14: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z6 @ Z14 )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( 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 ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z14 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% ms_weakI2
thf(fact_6502_pw__leq__split,axiom,
! [X7: multiset @ ( product_prod @ nat @ nat ),Y4: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ X7 @ Y4 )
=> ? [A10: multiset @ ( product_prod @ nat @ nat ),B10: multiset @ ( product_prod @ nat @ nat ),Z11: multiset @ ( product_prod @ nat @ nat )] :
( ( X7
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A10 @ Z11 ) )
& ( Y4
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B10 @ Z11 ) )
& ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A10 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B10 ) ) @ fun_max_strict )
| ( ( B10
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A10
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% pw_leq_split
thf(fact_6503_multp__code__def,axiom,
! [A: $tType] :
( ( multp_code @ A )
= ( ^ [P2: A > A > $o,N11: multiset @ A,M9: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M9 @ ( inter_mset @ A @ M9 @ N11 ) )
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ N11 @ ( inter_mset @ A @ M9 @ N11 ) ) ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M9 @ ( inter_mset @ A @ M9 @ N11 ) ) ) )
& ( P2 @ X3 @ Y3 ) ) ) ) ) ) ).
% multp_code_def
thf(fact_6504_count__image__mset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: multiset @ B,X: A] :
( ( count @ A @ ( image_mset @ B @ A @ F2 @ A4 ) @ X )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ A4 ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( set_mset @ B @ A4 ) ) ) ) ).
% count_image_mset
thf(fact_6505_set__mset__inter,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( set_mset @ A @ ( inter_mset @ A @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( set_mset @ A @ A4 ) @ ( set_mset @ A @ B3 ) ) ) ).
% set_mset_inter
thf(fact_6506_count__greater__eq__one__iff,axiom,
! [A: $tType,M4: multiset @ A,X: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( count @ A @ M4 @ X ) )
= ( member @ A @ X @ ( set_mset @ A @ M4 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_6507_count__mset__set_I1_J,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
= ( one_one @ nat ) ) ) ) ).
% count_mset_set(1)
thf(fact_6508_count__sum,axiom,
! [A: $tType,B: $tType,F2: B > ( multiset @ A ),A4: set @ B,X: A] :
( ( count @ A @ ( groups7311177749621191930dd_sum @ B @ ( multiset @ A ) @ F2 @ A4 ) @ X )
= ( groups7311177749621191930dd_sum @ B @ nat
@ ^ [A5: B] : ( count @ A @ ( F2 @ A5 ) @ X )
@ A4 ) ) ).
% count_sum
thf(fact_6509_set__mset__def,axiom,
! [A: $tType] :
( ( set_mset @ A )
= ( ^ [M9: multiset @ A] :
( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ M9 @ X3 ) ) ) ) ) ).
% set_mset_def
thf(fact_6510_set__mset__diff,axiom,
! [A: $tType,M4: multiset @ A,N4: multiset @ A] :
( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ N4 ) )
= ( collect @ A
@ ^ [A5: A] : ( ord_less @ nat @ ( count @ A @ N4 @ A5 ) @ ( count @ A @ M4 @ A5 ) ) ) ) ).
% set_mset_diff
thf(fact_6511_count__mset,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( count @ A @ ( mset @ A @ Xs ) @ X )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Xs ) ) ) ).
% count_mset
thf(fact_6512_count__image__mset_H,axiom,
! [A: $tType,B: $tType,F2: B > A,X7: multiset @ B,Y: A] :
( ( count @ A @ ( image_mset @ B @ A @ F2 @ X7 ) @ Y )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ X7 )
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ ( set_mset @ B @ X7 ) )
& ( Y
= ( F2 @ X3 ) ) ) ) ) ) ).
% count_image_mset'
thf(fact_6513_count__induct,axiom,
! [A: $tType,Y: A > nat,P: ( A > nat ) > $o] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
=> ( ! [X2: multiset @ A] : ( P @ ( count @ A @ X2 ) )
=> ( P @ Y ) ) ) ).
% count_induct
thf(fact_6514_count__cases,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
=> ~ ! [X2: multiset @ A] :
( Y
!= ( count @ A @ X2 ) ) ) ).
% count_cases
thf(fact_6515_count,axiom,
! [A: $tType,X: multiset @ A] :
( member @ ( A > nat ) @ ( count @ A @ X )
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) ) ).
% count
thf(fact_6516_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X: set @ ( multiset @ A )] :
( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ X ) )
= ( ^ [I3: A] :
( if @ nat
@ ( ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X )
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I3 )
@ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X ) ) ) ) ) ) ).
% Inf_multiset.rep_eq
thf(fact_6517_count__Inf__multiset__nonempty,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: A] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) @ X )
= ( complete_Inf_Inf @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X4: multiset @ A] : ( count @ A @ X4 @ X )
@ A4 ) ) ) ) ).
% count_Inf_multiset_nonempty
thf(fact_6518_count__single,axiom,
! [A: $tType,B2: A,A3: A] :
( ( ( B2 = A3 )
=> ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
= ( one_one @ nat ) ) )
& ( ( B2 != A3 )
=> ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ).
% count_single
thf(fact_6519_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( member @ A @ A3 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A3 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A3 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_6520_count__mset__set_H,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) )
=> ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) )
=> ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
= ( zero_zero @ nat ) ) ) ) ).
% count_mset_set'
thf(fact_6521_fold__mset__def,axiom,
! [B: $tType,A: $tType] :
( ( fold_mset @ A @ B )
= ( ^ [F: A > B > B,S2: B,M9: multiset @ A] :
( finite_fold @ A @ B
@ ^ [X3: A] : ( compow @ ( B > B ) @ ( count @ A @ M9 @ X3 ) @ ( F @ X3 ) )
@ S2
@ ( set_mset @ A @ M9 ) ) ) ) ).
% fold_mset_def
thf(fact_6522_distinct__count__atmost__1,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [X3: list @ A] :
! [A5: A] :
( ( ( member @ A @ A5 @ ( set2 @ A @ X3 ) )
=> ( ( count @ A @ ( mset @ A @ X3 ) @ A5 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A5 @ ( set2 @ A @ X3 ) )
=> ( ( count @ A @ ( mset @ A @ X3 ) @ A5 )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_6523_mult__cancel__max0,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X7: multiset @ A,Y4: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X7 @ Y4 ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X7 @ ( inter_mset @ A @ X7 @ Y4 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y4 @ ( inter_mset @ A @ X7 @ Y4 ) ) ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_max0
thf(fact_6524_multeqp__code__def,axiom,
! [A: $tType] :
( ( multeqp_code @ A )
= ( ^ [P2: A > A > $o,N11: multiset @ A,M9: multiset @ A] :
! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ N11 @ ( inter_mset @ A @ M9 @ N11 ) ) ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M9 @ ( inter_mset @ A @ M9 @ N11 ) ) ) )
& ( P2 @ X3 @ Y3 ) ) ) ) ) ).
% multeqp_code_def
thf(fact_6525_size__multiset__eq,axiom,
! [A: $tType] :
( ( size_multiset @ A )
= ( ^ [F: A > nat,M9: multiset @ A] :
( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( times_times @ nat @ ( count @ A @ M9 @ X3 ) @ ( suc @ ( F @ X3 ) ) )
@ ( set_mset @ A @ M9 ) ) ) ) ).
% size_multiset_eq
thf(fact_6526_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_6527_Inf__multiset__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( multiset @ A ) )
= ( map_fun @ ( set @ ( multiset @ A ) ) @ ( set @ ( A > nat ) ) @ ( A > nat ) @ ( multiset @ A ) @ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) ) @ ( abs_multiset @ A )
@ ^ [A6: set @ ( A > nat ),I3: A] :
( if @ nat
@ ( A6
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I3 )
@ A6 ) ) ) ) ) ).
% Inf_multiset_def
thf(fact_6528_count__Abs__multiset,axiom,
! [A: $tType,F2: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ F2 ) )
= F2 ) ) ).
% count_Abs_multiset
thf(fact_6529_zero__multiset__def,axiom,
! [A: $tType] :
( ( zero_zero @ ( multiset @ A ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( zero_zero @ nat ) ) ) ).
% zero_multiset_def
thf(fact_6530_Abs__multiset__inject,axiom,
! [A: $tType,X: A > nat,Y: A > nat] :
( ( member @ ( A > nat ) @ X
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
=> ( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
=> ( ( ( abs_multiset @ A @ X )
= ( abs_multiset @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_multiset_inject
thf(fact_6531_Abs__multiset__induct,axiom,
! [A: $tType,P: ( multiset @ A ) > $o,X: multiset @ A] :
( ! [Y2: A > nat] :
( ( member @ ( A > nat ) @ Y2
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
=> ( P @ ( abs_multiset @ A @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_multiset_induct
thf(fact_6532_Abs__multiset__cases,axiom,
! [A: $tType,X: multiset @ A] :
~ ! [Y2: A > nat] :
( ( X
= ( abs_multiset @ A @ Y2 ) )
=> ~ ( member @ ( A > nat ) @ Y2
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% Abs_multiset_cases
thf(fact_6533_plus__multiset__def,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( plus_plus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) ) ) ) ).
% plus_multiset_def
thf(fact_6534_minus__multiset__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( multiset @ A ) )
= ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( minus_minus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) ) ) ) ).
% minus_multiset_def
thf(fact_6535_Abs__multiset__inverse,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ Y ) )
= Y ) ) ).
% Abs_multiset_inverse
thf(fact_6536_type__definition__multiset,axiom,
! [A: $tType] :
( type_definition @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ ( abs_multiset @ A )
@ ( collect @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) ) ).
% type_definition_multiset
thf(fact_6537_add__mset__def,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( map_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [A5: A,M9: A > nat,B4: A] : ( if @ nat @ ( B4 = A5 ) @ ( suc @ ( M9 @ B4 ) ) @ ( M9 @ B4 ) ) ) ) ).
% add_mset_def
thf(fact_6538_size__Diff__singleton,axiom,
! [A: $tType,X: A,M4: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M4 ) @ ( one_one @ nat ) ) ) ) ).
% size_Diff_singleton
thf(fact_6539_size__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: multiset @ A] :
( ( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set_mset @ A @ A4 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( size_size @ ( multiset @ A ) @ A4 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_6540_image__mset__const__eq,axiom,
! [B: $tType,A: $tType,C2: A,M4: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [A5: B] : C2
@ M4 )
= ( replicate_mset @ A @ ( size_size @ ( multiset @ B ) @ M4 ) @ C2 ) ) ).
% image_mset_const_eq
thf(fact_6541_size__multiset__overloaded__def,axiom,
! [B: $tType] :
( ( size_size @ ( multiset @ B ) )
= ( size_multiset @ B
@ ^ [Uu: B] : ( zero_zero @ nat ) ) ) ).
% size_multiset_overloaded_def
thf(fact_6542_size__1__singleton__mset,axiom,
! [A: $tType,M4: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M4 )
= ( one_one @ nat ) )
=> ? [A8: A] :
( M4
= ( add_mset @ A @ A8 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% size_1_singleton_mset
thf(fact_6543_size__single,axiom,
! [A: $tType,B2: A] :
( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( one_one @ nat ) ) ).
% size_single
thf(fact_6544_size__multiset__overloaded__eq,axiom,
! [B: $tType] :
( ( size_size @ ( multiset @ B ) )
= ( ^ [X3: multiset @ B] : ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ X3 ) @ ( set_mset @ B @ X3 ) ) ) ) ).
% size_multiset_overloaded_eq
thf(fact_6545_mset__size1elem,axiom,
! [A: $tType,P: multiset @ A,Q4: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
=> ( ( member @ A @ Q4 @ ( set_mset @ A @ P ) )
=> ( P
= ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size1elem
thf(fact_6546_size__diff__se,axiom,
! [A: $tType,T4: A,S: multiset @ A] :
( ( member @ A @ T4 @ ( set_mset @ A @ S ) )
=> ( ( size_size @ ( multiset @ A ) @ S )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_6547_repeat__mset__def,axiom,
! [A: $tType] :
( ( repeat_mset @ A )
= ( map_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ nat ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [N2: nat,M9: A > nat,A5: A] : ( times_times @ nat @ N2 @ ( M9 @ A5 ) ) ) ) ).
% repeat_mset_def
thf(fact_6548_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [Y: B,A4: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [X3: A] : Y
@ A4 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( multiset @ A ) @ A4 ) ) @ Y ) ) ) ).
% sum_mset_constant
thf(fact_6549_Union__image__single__mset,axiom,
! [A: $tType,M: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ ( multiset @ A )
@ ( image_mset @ A @ ( multiset @ A )
@ ^ [X3: A] : ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) )
@ M ) )
= M ) ).
% Union_image_single_mset
thf(fact_6550_sum__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A4 ) )
= ( zero_zero @ A ) ) ) ).
% sum_mset.neutral_const
thf(fact_6551_set__mset__Union__mset,axiom,
! [A: $tType,MM: multiset @ ( multiset @ A )] :
( ( set_mset @ A @ ( comm_m7189776963980413722m_mset @ ( multiset @ A ) @ MM ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ ( set_mset @ ( multiset @ A ) @ MM ) ) ) ) ).
% set_mset_Union_mset
thf(fact_6552_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,A3: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N @ A3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ A3 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_6553_sum__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > C > A,B3: multiset @ C,A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [I3: B] : ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ C @ A @ ( G2 @ I3 ) @ B3 ) )
@ A4 ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [I3: B] : ( G2 @ I3 @ J3 )
@ A4 ) )
@ B3 ) ) ) ) ).
% sum_mset.swap
thf(fact_6554_sum__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,H3: B > A,A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A4 ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G2 @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ H3 @ A4 ) ) ) ) ) ).
% sum_mset.distrib
thf(fact_6555_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F2: B > A,M4: multiset @ B] :
( ( times_times @ A @ C2 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F2 @ M4 ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C2 @ ( F2 @ X3 ) )
@ M4 ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_6556_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,M4: multiset @ B,C2: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F2 @ M4 ) ) @ C2 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ C2 )
@ M4 ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_6557_sum__mset__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( times @ A )
& ( semiring_0 @ B ) )
=> ! [F2: A > B,A4: multiset @ A,G2: C > B,B3: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F2 @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G2 @ B3 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I3: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I3 ) @ ( G2 @ J3 ) )
@ B3 ) )
@ A4 ) ) ) ) ).
% sum_mset_product
thf(fact_6558_size__eq__sum__mset,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) )
= ( ^ [M9: multiset @ A] :
( comm_m7189776963980413722m_mset @ nat
@ ( image_mset @ A @ nat
@ ^ [A5: A] : ( one_one @ nat )
@ M9 ) ) ) ) ).
% size_eq_sum_mset
thf(fact_6559_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C2 @ ( zero_zero @ A ) )
@ A4 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A4 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_6560_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C2 @ ( zero_zero @ A ) )
@ A4 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A4 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_6561_repeat__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ X
@ X )
=> ( ( repeat_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( times_times @ nat @ Xa @ ( X @ A5 ) ) ) ) ) ).
% repeat_mset.abs_eq
thf(fact_6562_filter__mset__def,axiom,
! [A: $tType] :
( ( filter_mset @ A )
= ( map_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ ( A > $o ) ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [P2: A > $o,M9: A > nat,X3: A] : ( if @ nat @ ( P2 @ X3 ) @ ( M9 @ X3 ) @ ( zero_zero @ nat ) ) ) ) ).
% filter_mset_def
thf(fact_6563_filter__mset__True,axiom,
! [A: $tType,M4: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y3: A] : $true
@ M4 )
= M4 ) ).
% filter_mset_True
thf(fact_6564_filter__mset__False,axiom,
! [A: $tType,M4: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y3: A] : $false
@ M4 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% filter_mset_False
thf(fact_6565_set__mset__filter,axiom,
! [A: $tType,P: A > $o,M4: multiset @ A] :
( ( set_mset @ A @ ( filter_mset @ A @ P @ M4 ) )
= ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ ( set_mset @ A @ M4 ) )
& ( P @ A5 ) ) ) ) ).
% set_mset_filter
thf(fact_6566_filter__mset__mset__set,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( filter_mset @ A @ P @ ( mset_set @ A @ A4 ) )
= ( mset_set @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% filter_mset_mset_set
thf(fact_6567_mset__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( mset @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter_mset @ A @ P @ ( mset @ A @ Xs ) ) ) ).
% mset_filter
thf(fact_6568_filter__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: A > $o] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ X
@ X )
=> ( ( filter_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [X3: A] : ( if @ nat @ ( Xa @ X3 ) @ ( X @ X3 ) @ ( zero_zero @ nat ) ) ) ) ) ).
% filter_mset.abs_eq
thf(fact_6569_multiset__partition,axiom,
! [A: $tType,M4: multiset @ A,P: A > $o] :
( M4
= ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M4 )
@ ( filter_mset @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ M4 ) ) ) ).
% multiset_partition
thf(fact_6570_rel__fun__eq__eq__onp,axiom,
! [B: $tType,A: $tType,P: B > $o] :
( ( bNF_rel_fun @ A @ A @ B @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_eq_onp @ B @ P ) )
= ( bNF_eq_onp @ ( A > B )
@ ^ [F: A > B] :
! [X3: A] : ( P @ ( F @ X3 ) ) ) ) ).
% rel_fun_eq_eq_onp
thf(fact_6571_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R2: A > $o,X3: A,Y3: A] :
( ( R2 @ X3 )
& ( X3 = Y3 ) ) ) ) ).
% eq_onp_def
thf(fact_6572_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu: A] : $true )
= ( ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% eq_onp_True
thf(fact_6573_filter__filter__mset,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,M4: multiset @ A] :
( ( filter_mset @ A @ P @ ( filter_mset @ A @ Q2 @ M4 ) )
= ( filter_mset @ A
@ ^ [X3: A] :
( ( Q2 @ X3 )
& ( P @ X3 ) )
@ M4 ) ) ).
% filter_filter_mset
thf(fact_6574_eq__onp__top__eq__eq,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
= ( ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% eq_onp_top_eq_eq
thf(fact_6575_Quotient__crel__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T2 )
=> ( T2
= ( ^ [X3: A,Y3: B] :
( X3
= ( Rep @ Y3 ) ) ) ) ) ).
% Quotient_crel_typedef
thf(fact_6576_Quotient__eq__onp__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,Cr: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ Cr )
=> ( type_definition @ B @ A @ Rep @ Abs @ ( collect @ A @ P ) ) ) ).
% Quotient_eq_onp_typedef
thf(fact_6577_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o,T2: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( ( T2
= ( ^ [X3: B,Y3: A] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( quotient @ B @ A @ ( bNF_eq_onp @ B @ P ) @ Abs @ Rep @ T2 ) ) ) ).
% open_typedef_to_Quotient
thf(fact_6578_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: set @ B,T2: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ S )
=> ( ( T2
= ( ^ [X3: B,Y3: A] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( quotient @ B @ A
@ ( bNF_eq_onp @ B
@ ^ [X3: B] : ( member @ B @ X3 @ S ) )
@ Abs
@ Rep
@ T2 ) ) ) ).
% typedef_to_Quotient
thf(fact_6579_image__mset__If,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: B > A,G2: B > A,A4: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 )
= ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F2 @ ( filter_mset @ B @ P @ A4 ) )
@ ( image_mset @ B @ A @ G2
@ ( filter_mset @ B
@ ^ [X3: B] :
~ ( P @ X3 )
@ A4 ) ) ) ) ).
% image_mset_If
thf(fact_6580_rel__fun__eq__onp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R: A > $o,S: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C @ ( bNF_eq_onp @ A @ R ) @ S )
= ( ^ [F: A > B,G: A > C] :
! [X3: A] :
( ( R @ X3 )
=> ( S @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% rel_fun_eq_onp_rel
thf(fact_6581_filter__eq__replicate__mset,axiom,
! [A: $tType,X: A,D4: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y3: A] : Y3 = X
@ D4 )
= ( replicate_mset @ A @ ( count @ A @ D4 @ X ) @ X ) ) ).
% filter_eq_replicate_mset
thf(fact_6582_filter__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y5: A > $o,Z4: A > $o] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
@ ^ [P2: A > $o,M9: A > nat,X3: A] : ( if @ nat @ ( P2 @ X3 ) @ ( M9 @ X3 ) @ ( zero_zero @ nat ) )
@ ^ [P2: A > $o,M9: A > nat,X3: A] : ( if @ nat @ ( P2 @ X3 ) @ ( M9 @ X3 ) @ ( zero_zero @ nat ) ) ) ).
% filter_mset.rsp
thf(fact_6583_zero__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ ^ [A5: A] : ( zero_zero @ nat )
@ ^ [A5: A] : ( zero_zero @ nat ) ) ).
% zero_multiset.rsp
thf(fact_6584_add__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
@ ^ [A5: A,M9: A > nat,B4: A] : ( if @ nat @ ( B4 = A5 ) @ ( suc @ ( M9 @ B4 ) ) @ ( M9 @ B4 ) )
@ ^ [A5: A,M9: A > nat,B4: A] : ( if @ nat @ ( B4 = A5 ) @ ( suc @ ( M9 @ B4 ) ) @ ( M9 @ B4 ) ) ) ).
% add_mset.rsp
thf(fact_6585_plus__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( plus_plus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( plus_plus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) ) ) ).
% plus_multiset.rsp
thf(fact_6586_minus__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( minus_minus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( minus_minus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) ) ) ).
% minus_multiset.rsp
thf(fact_6587_repeat__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
@ ^ [N2: nat,M9: A > nat,A5: A] : ( times_times @ nat @ N2 @ ( M9 @ A5 ) )
@ ^ [N2: nat,M9: A > nat,A5: A] : ( times_times @ nat @ N2 @ ( M9 @ A5 ) ) ) ).
% repeat_mset.rsp
thf(fact_6588_add__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: A] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ X
@ X )
=> ( ( add_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [B4: A] : ( if @ nat @ ( B4 = Xa ) @ ( suc @ ( X @ B4 ) ) @ ( X @ B4 ) ) ) ) ) ).
% add_mset.abs_eq
thf(fact_6589_plus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ X
@ X )
=> ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( plus_plus @ nat @ ( Xa @ A5 ) @ ( X @ A5 ) ) ) ) ) ) ).
% plus_multiset.abs_eq
thf(fact_6590_minus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) )
@ X
@ X )
=> ( ( minus_minus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( minus_minus @ nat @ ( Xa @ A5 ) @ ( X @ A5 ) ) ) ) ) ) ).
% minus_multiset.abs_eq
thf(fact_6591_Quotient__multiset,axiom,
! [A: $tType] :
( quotient @ ( A > nat ) @ ( multiset @ A )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ( abs_multiset @ A )
@ ( count @ A )
@ ( cr_multiset @ A ) ) ).
% Quotient_multiset
thf(fact_6592_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X: set @ ( A > nat )] :
( ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ X
@ X )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ ( A > nat ) @ ( multiset @ A ) @ ( abs_multiset @ A ) @ X ) )
= ( abs_multiset @ A
@ ^ [I3: A] :
( if @ nat
@ ( X
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I3 )
@ X ) ) ) ) ) ) ).
% Inf_multiset.abs_eq
thf(fact_6593_rel__set__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_set @ A @ B )
= ( ^ [R2: A > B > $o,A6: set @ A,B5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ B5 )
& ( R2 @ X3 @ Y3 ) ) )
& ! [X3: B] :
( ( member @ B @ X3 @ B5 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( R2 @ Y3 @ X3 ) ) ) ) ) ) ).
% rel_set_def
thf(fact_6594_fun_Oset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R )
@ ( bNF_rel_set @ A @ B @ R )
@ ^ [F: D > A] : ( image2 @ D @ A @ F @ ( top_top @ ( set @ D ) ) )
@ ^ [F: D > B] : ( image2 @ D @ B @ F @ ( top_top @ ( set @ D ) ) ) ) ).
% fun.set_transfer
thf(fact_6595_cr__multiset__def,axiom,
! [A: $tType] :
( ( cr_multiset @ A )
= ( ^ [X3: A > nat,Y3: multiset @ A] :
( X3
= ( count @ A @ Y3 ) ) ) ) ).
% cr_multiset_def
thf(fact_6596_Inf__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( set @ ( A > nat ) ) @ ( set @ ( A > nat ) ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
@ ^ [A6: set @ ( A > nat ),I3: A] :
( if @ nat
@ ( A6
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I3 )
@ A6 ) ) )
@ ^ [A6: set @ ( A > nat ),I3: A] :
( if @ nat
@ ( A6
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I3 )
@ A6 ) ) ) ) ).
% Inf_multiset.rsp
thf(fact_6597_INF__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Inf @ C )
=> ! [A4: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ A4 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ A4
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [A6: set @ A,F: A > C] : ( complete_Inf_Inf @ C @ ( image2 @ A @ C @ F @ A6 ) )
@ ^ [A6: set @ B,F: B > C] : ( complete_Inf_Inf @ C @ ( image2 @ B @ C @ F @ A6 ) ) ) ) ).
% INF_parametric
thf(fact_6598_SUP__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Sup @ C )
=> ! [R: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ R )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ R
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [A6: set @ A,F: A > C] : ( complete_Sup_Sup @ C @ ( image2 @ A @ C @ F @ A6 ) )
@ ^ [A6: set @ B,F: B > C] : ( complete_Sup_Sup @ C @ ( image2 @ B @ C @ F @ A6 ) ) ) ) ).
% SUP_parametric
thf(fact_6599_empty__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_set @ A @ B @ A4 @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ B ) ) ) ).
% empty_transfer
thf(fact_6600_union__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_set @ A @ B @ A4 ) ) @ ( sup_sup @ ( set @ A ) ) @ ( sup_sup @ ( set @ B ) ) ) ).
% union_transfer
thf(fact_6601_Union__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) ) ).
% Union_transfer
thf(fact_6602_set__relator__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( bNF_rel_set @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( set @ A )
@ ^ [A6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( P @ X3 ) ) ) ) ).
% set_relator_eq_onp
thf(fact_6603_UNION__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B3: C > D > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > ( set @ C ) ) > ( set @ C ) ) @ ( ( B > ( set @ D ) ) > ( set @ D ) ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( set @ C ) ) @ ( B > ( set @ D ) ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_fun @ A @ B @ ( set @ C ) @ ( set @ D ) @ A4 @ ( bNF_rel_set @ C @ D @ B3 ) ) @ ( bNF_rel_set @ C @ D @ B3 ) )
@ ^ [A6: set @ A,F: A > ( set @ C )] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ F @ A6 ) )
@ ^ [A6: set @ B,F: B > ( set @ D )] : ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ B @ ( set @ D ) @ F @ A6 ) ) ) ).
% UNION_transfer
thf(fact_6604_Inf__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( set @ ( A > nat ) ) @ ( set @ ( multiset @ A ) ) @ ( A > nat ) @ ( multiset @ A )
@ ( bNF_rel_set @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ^ [A6: set @ ( A > nat ),I3: A] :
( if @ nat
@ ( A6
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I3 )
@ A6 ) ) )
@ ( complete_Inf_Inf @ ( multiset @ A ) ) ) ).
% Inf_multiset.transfer
thf(fact_6605_multp__def,axiom,
! [A: $tType] :
( ( multp @ A )
= ( ^ [R4: A > A > $o,M9: multiset @ A,N11: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M9 @ N11 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% multp_def
thf(fact_6606_zero__multiset_Otransfer,axiom,
! [A: $tType] :
( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [A5: A] : ( zero_zero @ nat )
@ ( zero_zero @ ( multiset @ A ) ) ) ).
% zero_multiset.transfer
thf(fact_6607_multiset_Orep__transfer,axiom,
! [D: $tType,E: $tType,T2: D > E > $o] :
( bNF_rel_fun @ ( D > nat ) @ ( multiset @ E ) @ ( D > nat ) @ ( E > nat ) @ ( pcr_multiset @ D @ E @ T2 )
@ ( bNF_rel_fun @ D @ E @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ^ [X3: D > nat] : X3
@ ( count @ E ) ) ).
% multiset.rep_transfer
thf(fact_6608_add__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [A5: A,M9: A > nat,B4: A] : ( if @ nat @ ( B4 = A5 ) @ ( suc @ ( M9 @ B4 ) ) @ ( M9 @ B4 ) )
@ ( add_mset @ A ) ) ).
% add_mset.transfer
thf(fact_6609_plus__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( plus_plus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) )
@ ( plus_plus @ ( multiset @ A ) ) ) ).
% plus_multiset.transfer
thf(fact_6610_minus__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [M9: A > nat,N11: A > nat,A5: A] : ( minus_minus @ nat @ ( M9 @ A5 ) @ ( N11 @ A5 ) )
@ ( minus_minus @ ( multiset @ A ) ) ) ).
% minus_multiset.transfer
thf(fact_6611_filter__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y5: A > $o,Z4: A > $o] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [P2: A > $o,M9: A > nat,X3: A] : ( if @ nat @ ( P2 @ X3 ) @ ( M9 @ X3 ) @ ( zero_zero @ nat ) )
@ ( filter_mset @ A ) ) ).
% filter_mset.transfer
thf(fact_6612_repeat__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [N2: nat,M9: A > nat,A5: A] : ( times_times @ nat @ N2 @ ( M9 @ A5 ) )
@ ( repeat_mset @ A ) ) ).
% repeat_mset.transfer
thf(fact_6613_multiset_Odomain,axiom,
! [B: $tType,C: $tType,T2: C > B > $o] :
( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T2 ) )
= ( ^ [X3: C > nat] :
? [Y3: B > nat] :
( ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ X3
@ Y3 )
& ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Z5: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( Y3 @ Z5 ) ) ) ) ) ) ) ).
% multiset.domain
thf(fact_6614_prod__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,A4: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A4 )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% prod_mset.remove
thf(fact_6615_prod__mset__empty,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( one_one @ A ) ) ) ).
% prod_mset_empty
thf(fact_6616_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,N4: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( add_mset @ A @ X @ N4 ) )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ N4 ) ) ) ) ).
% prod_mset.add_mset
thf(fact_6617_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ A,B3: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B3 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( comm_m9189036328036947845d_mset @ A @ B3 ) ) ) ) ).
% prod_mset_Un
thf(fact_6618_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M4: multiset @ A,N4: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M4 @ N4 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M4 ) @ ( comm_m9189036328036947845d_mset @ A @ N4 ) ) ) ) ).
% prod_mset.union
thf(fact_6619_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A4 ) )
= ( one_one @ A ) ) ) ).
% prod_mset.neutral_const
thf(fact_6620_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,X: B,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ ( add_mset @ B @ X @ A4 ) ) )
= ( times_times @ A @ ( G2 @ X ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% prod_mset.insert
thf(fact_6621_prod__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : C2
@ A4 ) )
= ( power_power @ A @ C2 @ ( size_size @ ( multiset @ B ) @ A4 ) ) ) ) ).
% prod_mset_constant
thf(fact_6622_Domain__eq__top,axiom,
! [A: $tType] :
( ( domainp @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( top_top @ ( A > $o ) ) ) ).
% Domain_eq_top
thf(fact_6623_pcr__Domainp,axiom,
! [B: $tType,A: $tType,C: $tType,B3: A > B > $o,P: A > $o,A4: C > A > $o] :
( ( ( domainp @ A @ B @ B3 )
= P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A4 @ B3 ) )
= ( ^ [X3: C] :
? [Y3: A] :
( ( A4 @ X3 @ Y3 )
& ( P @ Y3 ) ) ) ) ) ).
% pcr_Domainp
thf(fact_6624_prod__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > C > A,B3: multiset @ C,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [I3: B] : ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ C @ A @ ( G2 @ I3 ) @ B3 ) )
@ A4 ) )
= ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [I3: B] : ( G2 @ I3 @ J3 )
@ A4 ) )
@ B3 ) ) ) ) ).
% prod_mset.swap
thf(fact_6625_prod__mset_Oneutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ A4 ) )
=> ( X2
= ( one_one @ A ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod_mset.neutral
thf(fact_6626_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,H3: B > A,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A4 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ A4 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H3 @ A4 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_6627_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: multiset @ A] :
( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( one_one @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ A4 ) )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% is_unit_prod_mset_iff
thf(fact_6628_pcr__Domainp__par,axiom,
! [A: $tType,B: $tType,C: $tType,B3: A > B > $o,P23: A > $o,A4: C > A > $o,P12: C > $o,P24: C > $o] :
( ( ( domainp @ A @ B @ B3 )
= P23 )
=> ( ( ( domainp @ C @ A @ A4 )
= P12 )
=> ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P24
@ P23 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A4 @ B3 ) )
= ( inf_inf @ ( C > $o ) @ P12 @ P24 ) ) ) ) ) ).
% pcr_Domainp_par
thf(fact_6629_prod__mset_Oeq__fold,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( fold_mset @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_mset.eq_fold
thf(fact_6630_multiset_Odomain__eq,axiom,
! [A: $tType] :
( ( domainp @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
= ( ^ [F: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ) ) ).
% multiset.domain_eq
thf(fact_6631_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C2 @ ( one_one @ A ) )
@ A4 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_6632_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C2 @ ( one_one @ A ) )
@ A4 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_6633_prod__mset__multiplicity,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( ^ [M9: multiset @ A] :
( groups7121269368397514597t_prod @ A @ A
@ ^ [X3: A] : ( power_power @ A @ X3 @ ( count @ A @ M9 @ X3 ) )
@ ( set_mset @ A @ M9 ) ) ) ) ) ).
% prod_mset_multiplicity
thf(fact_6634_multiset_Odomain__par__left__total,axiom,
! [B: $tType,C: $tType,T2: C > B > $o,P7: ( C > nat ) > $o] :
( ( left_total @ ( C > nat ) @ ( B > nat )
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 ) )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P7
@ ^ [F: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T2 ) )
= P7 ) ) ) ).
% multiset.domain_par_left_total
thf(fact_6635_composed__equiv__rel__eq__onp,axiom,
! [B: $tType,A: $tType,R: A > B > $o,P: A > $o,P7: B > $o,P10: A > $o] :
( ( left_unique @ A @ B @ R )
=> ( ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P
@ P7 )
=> ( ( ( domainp @ A @ B @ R )
= P10 )
=> ( ( relcompp @ A @ B @ A @ R @ ( relcompp @ B @ B @ A @ ( bNF_eq_onp @ B @ P7 ) @ ( conversep @ A @ B @ R ) ) )
= ( bNF_eq_onp @ A @ ( inf_inf @ ( A > $o ) @ P10 @ P ) ) ) ) ) ) ).
% composed_equiv_rel_eq_onp
thf(fact_6636_typedef__left__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( T2
= ( ^ [X3: A,Y3: B] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( left_unique @ A @ B @ T2 ) ) ) ).
% typedef_left_unique
thf(fact_6637_left__unique__iff,axiom,
! [B: $tType,A: $tType] :
( ( left_unique @ A @ B )
= ( ^ [R2: A > B > $o] :
! [Z5: B] :
( uniq @ A
@ ^ [X3: A] : ( R2 @ X3 @ Z5 ) ) ) ) ).
% left_unique_iff
thf(fact_6638_rat_Odomain,axiom,
( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( ^ [X3: product_prod @ int @ int] :
? [Y3: product_prod @ int @ int] :
( ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ X3
@ Y3 )
& ( ratrel @ Y3 @ Y3 ) ) ) ) ).
% rat.domain
thf(fact_6639_multiset_Odomain__par,axiom,
! [B: $tType,C: $tType,T2: C > B > $o,DT: C > $o,DS: nat > $o,P24: ( C > nat ) > $o] :
( ( ( domainp @ C @ B @ T2 )
= DT )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= DS )
=> ( ( left_unique @ C @ B @ T2 )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P24
@ ^ [F: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T2 ) )
= ( inf_inf @ ( ( C > nat ) > $o ) @ ( basic_pred_fun @ C @ nat @ DT @ DS ) @ P24 ) ) ) ) ) ) ).
% multiset.domain_par
thf(fact_6640_right__total__relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F4: $tType,B: $tType,D: $tType,B3: A > B > $o,A4: C > D > $o,C3: E > F4 > $o] :
( ( right_total @ A @ B @ B3 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F4 > $o ) > D > F4 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B3
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F4 > $o ) @ ( C > E > $o ) @ ( D > F4 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F4 > $o ) @ B3
@ ( bNF_rel_fun @ E @ F4 @ $o @ $o @ C3
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F4 > $o ) @ A4
@ ( bNF_rel_fun @ E @ F4 @ $o @ $o @ C3
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) ) )
@ ^ [R2: C > A > $o,S8: A > E > $o,X3: C,Z5: E] :
? [Y3: A] :
( ( member @ A @ Y3 @ ( collect @ A @ ( domainp @ A @ B @ B3 ) ) )
& ( R2 @ X3 @ Y3 )
& ( S8 @ Y3 @ Z5 ) )
@ ( relcompp @ D @ B @ F4 ) ) ) ).
% right_total_relcompp_transfer
thf(fact_6641_pred__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_fun @ A @ B )
= ( ^ [A6: A > $o,B5: B > $o,F: A > B] :
! [X3: A] :
( ( A6 @ X3 )
=> ( B5 @ ( F @ X3 ) ) ) ) ) ).
% pred_fun_def
thf(fact_6642_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F2: A > B,G2: A > B] :
( ( X = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z5: A] :
( ( F2 @ Z5 )
= ( G2 @ Z5 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F2 @ X )
= ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_6643_fun_Opred__True,axiom,
! [A: $tType,D: $tType] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Uu: A] : $true )
= ( ^ [Uu: D > A] : $true ) ) ).
% fun.pred_True
thf(fact_6644_fun_Opred__mono,axiom,
! [D: $tType,A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( D > A ) > $o )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa ) ) ) ).
% fun.pred_mono
thf(fact_6645_typedef__right__total,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( T2
= ( ^ [X3: A,Y3: B] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( right_total @ A @ B @ T2 ) ) ) ).
% typedef_right_total
thf(fact_6646_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P4: B > $o,F2: C > B] :
( ( nO_MATCH @ ( A > A ) @ ( B > $o ) @ ( id @ A ) @ P4 )
=> ( ( basic_pred_fun @ C @ B
@ ^ [X3: C] : $true
@ P4
@ F2 )
= ( basic_pred_fun @ C @ $o
@ ^ [X3: C] : $true
@ ( id @ $o )
@ ( comp @ B @ $o @ C @ P4 @ F2 ) ) ) ) ).
% pred_fun_True_id
thf(fact_6647_right__total__UNIV__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( right_total @ A @ B @ A4 )
=> ( bNF_rel_set @ A @ B @ A4 @ ( collect @ A @ ( domainp @ A @ B @ A4 ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% right_total_UNIV_transfer
thf(fact_6648_fun_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P: A > $o,X: D > A,Pa: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_6649_fun_Opred__cong,axiom,
! [A: $tType,D: $tType,X: D > A,Ya: D > A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_6650_fun_Opred__rel,axiom,
! [A: $tType,D: $tType,P: A > $o,X: D > A] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X )
= ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_eq_onp @ A @ P )
@ X
@ X ) ) ).
% fun.pred_rel
thf(fact_6651_fun_ODomainp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R: A > B > $o] :
( ( domainp @ ( C > A ) @ ( C > B )
@ ( bNF_rel_fun @ C @ C @ A @ B
@ ^ [Y5: C,Z4: C] : Y5 = Z4
@ R ) )
= ( basic_pred_fun @ C @ A
@ ^ [Uu: C] : $true
@ ( domainp @ A @ B @ R ) ) ) ).
% fun.Domainp_rel
thf(fact_6652_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q2: B > $o,F2: A > B,X: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q2
@ ( comp @ A @ B @ D @ F2 @ X ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q2 @ F2 )
@ X ) ) ).
% fun.pred_map
thf(fact_6653_right__total__Collect__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( right_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_set @ A @ B @ A4 )
@ ^ [P2: A > $o] :
( collect @ A
@ ^ [X3: A] :
( ( P2 @ X3 )
& ( domainp @ A @ B @ A4 @ X3 ) ) )
@ ( collect @ B ) ) ) ).
% right_total_Collect_transfer
thf(fact_6654_fun_Opred__set,axiom,
! [A: $tType,D: $tType,P: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P )
= ( ^ [X3: D > A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
=> ( P @ Y3 ) ) ) ) ).
% fun.pred_set
thf(fact_6655_fun_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true )
@ ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true ) ) ).
% fun.pred_transfer
thf(fact_6656_fun_Orel__eq__onp,axiom,
! [D: $tType,A: $tType,P: A > $o] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( D > A )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P ) ) ) ).
% fun.rel_eq_onp
thf(fact_6657_right__total__Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B3: A > B > $o,A4: C > D > $o] :
( ( right_total @ A @ B @ B3 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B3
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ^ [T7: C > A > $o,X3: C] :
? [Y3: A] :
( ( member @ A @ Y3 @ ( collect @ A @ ( domainp @ A @ B @ B3 ) ) )
& ( T7 @ X3 @ Y3 ) )
@ ( domainp @ D @ B ) ) ) ).
% right_total_Domainp_transfer
thf(fact_6658_rat_Odomain__par,axiom,
! [DR1: int > $o,DR2: int > $o,P24: ( product_prod @ int @ int ) > $o] :
( ( ( domainp @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
= DR1 )
=> ( ( ( domainp @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
= DR2 )
=> ( ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P24
@ ^ [X3: product_prod @ int @ int] : ( ratrel @ X3 @ X3 ) )
=> ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( inf_inf @ ( ( product_prod @ int @ int ) > $o ) @ ( basic_pred_prod @ int @ int @ DR1 @ DR2 ) @ P24 ) ) ) ) ) ).
% rat.domain_par
thf(fact_6659_rat_Odomain__par__left__total,axiom,
! [P7: ( product_prod @ int @ int ) > $o] :
( ( left_total @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 ) )
=> ( ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P7
@ ^ [X3: product_prod @ int @ int] : ( ratrel @ X3 @ X3 ) )
=> ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= P7 ) ) ) ).
% rat.domain_par_left_total
thf(fact_6660_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P12: A > $o,P23: B > $o,A3: A,B2: B] :
( ( basic_pred_prod @ A @ B @ P12 @ P23 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( ( P12 @ A3 )
& ( P23 @ B2 ) ) ) ).
% pred_prod_inject
thf(fact_6661_pred__prod__split,axiom,
! [B: $tType,A: $tType,P: $o > $o,Q2: A > $o,R: B > $o,Xy2: product_prod @ A @ B] :
( ( P @ ( basic_pred_prod @ A @ B @ Q2 @ R @ Xy2 ) )
= ( ! [X3: A,Y3: B] :
( ( Xy2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( P
@ ( ( Q2 @ X3 )
& ( R @ Y3 ) ) ) ) ) ) ).
% pred_prod_split
thf(fact_6662_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P12: A > $o,A3: A,P23: B > $o,B2: B] :
( ( P12 @ A3 )
=> ( ( P23 @ B2 )
=> ( basic_pred_prod @ A @ B @ P12 @ P23 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% pred_prod.intros
thf(fact_6663_pred__prod_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P13: A > $o,P25: B > $o,A5: product_prod @ A @ B] :
? [B4: A,C5: B] :
( ( A5
= ( product_Pair @ A @ B @ B4 @ C5 ) )
& ( P13 @ B4 )
& ( P25 @ C5 ) ) ) ) ).
% pred_prod.simps
thf(fact_6664_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P12: A > $o,P23: B > $o,A3: product_prod @ A @ B] :
( ( basic_pred_prod @ A @ B @ P12 @ P23 @ A3 )
=> ~ ! [A8: A,B7: B] :
( ( A3
= ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( ( P12 @ A8 )
=> ~ ( P23 @ B7 ) ) ) ) ).
% pred_prod.cases
thf(fact_6665_prod_Omap__cong__pred,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod @ A @ B,Ya: product_prod @ A @ B,F1: A > C,G1: A > C,F22: B > D,G22: B > D] :
( ( X = Ya )
=> ( ( basic_pred_prod @ A @ B
@ ^ [Z1: A] :
( ( F1 @ Z1 )
= ( G1 @ Z1 ) )
@ ^ [Z22: B] :
( ( F22 @ Z22 )
= ( G22 @ Z22 ) )
@ Ya )
=> ( ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ X )
= ( product_map_prod @ A @ C @ B @ D @ G1 @ G22 @ Ya ) ) ) ) ).
% prod.map_cong_pred
thf(fact_6666_prod_Opred__True,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B
@ ^ [Uu: A] : $true
@ ^ [Uu: B] : $true )
= ( ^ [Uu: product_prod @ A @ B] : $true ) ) ).
% prod.pred_True
thf(fact_6667_right__total__Inter__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( right_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) ) @ ( bNF_rel_set @ A @ B @ A4 )
@ ^ [S8: set @ ( set @ A )] : ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ S8 ) @ ( collect @ A @ ( domainp @ A @ B @ A4 ) ) )
@ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% right_total_Inter_transfer
thf(fact_6668_vimage__right__total__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B3: A > B > $o,A4: C > D > $o] :
( ( bi_unique @ A @ B @ B3 )
=> ( ( right_total @ C @ D @ A4 )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( set @ A ) > ( set @ C ) ) @ ( ( set @ B ) > ( set @ D ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ A4 @ B3 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_set @ A @ B @ B3 ) @ ( bNF_rel_set @ C @ D @ A4 ) )
@ ^ [F: C > A,X4: set @ A] : ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ X4 ) @ ( collect @ C @ ( domainp @ C @ D @ A4 ) ) )
@ ( vimage @ D @ B ) ) ) ) ).
% vimage_right_total_transfer
thf(fact_6669_bi__unique__iff,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [R2: A > B > $o] :
( ! [Z5: B] :
( uniq @ A
@ ^ [X3: A] : ( R2 @ X3 @ Z5 ) )
& ! [Z5: A] : ( uniq @ B @ ( R2 @ Z5 ) ) ) ) ) ).
% bi_unique_iff
thf(fact_6670_typedef__bi__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A4: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( T2
= ( ^ [X3: A,Y3: B] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( bi_unique @ A @ B @ T2 ) ) ) ).
% typedef_bi_unique
thf(fact_6671_inter__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_set @ A @ B @ A4 ) ) @ ( inf_inf @ ( set @ A ) ) @ ( inf_inf @ ( set @ B ) ) ) ) ).
% inter_transfer
thf(fact_6672_right__total__Compl__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( right_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_set @ A @ B @ A4 )
@ ^ [S8: set @ A] : ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ S8 ) @ ( collect @ A @ ( domainp @ A @ B @ A4 ) ) )
@ ( uminus_uminus @ ( set @ B ) ) ) ) ) ).
% right_total_Compl_transfer
thf(fact_6673_right__total__fun__eq__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B3: C > D > $o] :
( ( right_total @ A @ B @ A4 )
=> ( ( bi_unique @ C @ D @ B3 )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B3 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B3 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ^ [F: A > C,G: A > C] :
! [X3: A] :
( ( member @ A @ X3 @ ( collect @ A @ ( domainp @ A @ B @ A4 ) ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
@ ^ [Y5: B > D,Z4: B > D] : Y5 = Z4 ) ) ) ).
% right_total_fun_eq_transfer
thf(fact_6674_Inter__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) ) @ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% Inter_transfer
thf(fact_6675_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A4 ) @ ( rel_filter @ A @ B @ A4 ) ) @ ( inf_inf @ ( filter @ A ) ) @ ( inf_inf @ ( filter @ B ) ) ) ) ) ).
% inf_filter_parametric
thf(fact_6676_UNIV__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_set @ A @ B @ A4 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% UNIV_transfer
thf(fact_6677_top__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( rel_filter @ A @ B @ A4 @ ( top_top @ ( filter @ A ) ) @ ( top_top @ ( filter @ B ) ) ) ) ).
% top_filter_parametric
thf(fact_6678_Compl__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( bNF_rel_set @ A @ B @ A4 ) @ ( uminus_uminus @ ( set @ A ) ) @ ( uminus_uminus @ ( set @ B ) ) ) ) ) ).
% Compl_transfer
thf(fact_6679_Abs__rat__inject,axiom,
! [X: set @ ( product_prod @ int @ int ),Y: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ X
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ( ( abs_rat @ X )
= ( abs_rat @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_rat_inject
thf(fact_6680_Abs__rat__induct,axiom,
! [P: rat > $o,X: rat] :
( ! [Y2: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y2
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) )
=> ( P @ ( abs_rat @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_rat_induct
thf(fact_6681_Abs__rat__cases,axiom,
! [X: rat] :
~ ! [Y2: set @ ( product_prod @ int @ int )] :
( ( X
= ( abs_rat @ Y2 ) )
=> ~ ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y2
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) ) ) ).
% Abs_rat_cases
thf(fact_6682_type__definition__rat,axiom,
( type_definition @ rat @ ( set @ ( product_prod @ int @ int ) ) @ rep_rat @ abs_rat
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) ) ).
% type_definition_rat
thf(fact_6683_Abs__rat__inverse,axiom,
! [Y: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ( rep_rat @ ( abs_rat @ Y ) )
= Y ) ) ).
% Abs_rat_inverse
thf(fact_6684_Rep__rat__induct,axiom,
! [Y: set @ ( product_prod @ int @ int ),P: ( set @ ( product_prod @ int @ int ) ) > $o] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ! [X2: rat] : ( P @ ( rep_rat @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_rat_induct
thf(fact_6685_Rep__rat__cases,axiom,
! [Y: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) )
=> ~ ! [X2: rat] :
( Y
!= ( rep_rat @ X2 ) ) ) ).
% Rep_rat_cases
thf(fact_6686_Rep__rat,axiom,
! [X: rat] :
( member @ ( set @ ( product_prod @ int @ int ) ) @ ( rep_rat @ X )
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X3 ) ) ) ) ) ) ).
% Rep_rat
thf(fact_6687_wfP__SUP,axiom,
! [B: $tType,A: $tType,R3: A > B > B > $o] :
( ! [I2: A] : ( wfP @ B @ ( R3 @ I2 ) )
=> ( ! [I2: A,J2: A] :
( ( ( R3 @ I2 )
!= ( R3 @ J2 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R3 @ I2 ) ) @ ( rangep @ B @ B @ ( R3 @ J2 ) ) )
= ( bot_bot @ ( B > $o ) ) ) )
=> ( wfP @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% wfP_SUP
thf(fact_6688_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ B,B3: multiset @ B,G2: B > A] :
( ( ( inter_mset @ B @ A4 @ B3 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ ( union_mset @ B @ A4 @ B3 ) ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ A4 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ B3 ) ) ) ) ) ) ).
% prod_mset.union_disjoint
thf(fact_6689_wfP__empty,axiom,
! [A: $tType] :
( wfP @ A
@ ^ [X3: A,Y3: A] : $false ) ).
% wfP_empty
thf(fact_6690_set__mset__sup,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( set_mset @ A @ ( union_mset @ A @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ A4 ) @ ( set_mset @ A @ B3 ) ) ) ).
% set_mset_sup
thf(fact_6691_subset__mset_Ocomm__monoid__axioms,axiom,
! [A: $tType] : ( comm_monoid @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.comm_monoid_axioms
thf(fact_6692_subset__mset_Osemilattice__neutr__axioms,axiom,
! [A: $tType] : ( semilattice_neutr @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.semilattice_neutr_axioms
thf(fact_6693_subset__mset_Omonoid__axioms,axiom,
! [A: $tType] : ( monoid @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.monoid_axioms
thf(fact_6694_wfP__wf__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
= ( wf @ A @ R3 ) ) ).
% wfP_wf_eq
thf(fact_6695_wfP__def,axiom,
! [A: $tType] :
( ( wfP @ A )
= ( ^ [R4: A > A > $o] : ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ).
% wfP_def
thf(fact_6696_wfP__acyclicP,axiom,
! [A: $tType,R3: A > A > $o] :
( ( wfP @ A @ R3 )
=> ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ).
% wfP_acyclicP
thf(fact_6697_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A5: multiset @ A] :
( ( Uu
= ( inter_mset @ A @ X @ A5 ) )
& ( member @ ( multiset @ A ) @ A5 @ A4 ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup1_distrib
thf(fact_6698_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B3 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A5: multiset @ A,B4: multiset @ A] :
( ( Uu
= ( inter_mset @ A @ A5 @ B4 ) )
& ( member @ ( multiset @ A ) @ A5 @ A4 )
& ( member @ ( multiset @ A ) @ B4 @ B3 ) ) ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
thf(fact_6699_subset__mset_OSup__fin_Osingleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.Sup_fin.singleton
thf(fact_6700_subset__mset_OSup__fin_Oinsert,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.Sup_fin.insert
thf(fact_6701_subset__mset_OSup__fin_Osubset,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B3 @ A4 )
=> ( ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset
thf(fact_6702_subset__mset_OSup__fin_Ohom__commute,axiom,
! [A: $tType,H3: ( multiset @ A ) > ( multiset @ A ),N4: set @ ( multiset @ A )] :
( ! [X2: multiset @ A,Y2: multiset @ A] :
( ( H3 @ ( union_mset @ A @ X2 @ Y2 ) )
= ( union_mset @ A @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H3 @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ N4 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H3 @ N4 ) ) ) ) ) ) ).
% subset_mset.Sup_fin.hom_commute
thf(fact_6703_subset__mset_OSup__fin_Oeq__fold_H,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 )
= ( the2 @ ( multiset @ A )
@ ( finite_fold @ ( multiset @ A ) @ ( option @ ( multiset @ A ) )
@ ^ [X3: multiset @ A,Y3: option @ ( multiset @ A )] : ( some @ ( multiset @ A ) @ ( case_option @ ( multiset @ A ) @ ( multiset @ A ) @ X3 @ ( union_mset @ A @ X3 ) @ Y3 ) )
@ ( none @ ( multiset @ A ) )
@ A4 ) ) ) ).
% subset_mset.Sup_fin.eq_fold'
thf(fact_6704_subset__mset_OSup__fin_Ounion,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) )
= ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B3 ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.union
thf(fact_6705_subset__mset_OcSup__eq__Sup__fin,axiom,
! [A: $tType,X7: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X7 )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ X7 ) ) ) ) ).
% subset_mset.cSup_eq_Sup_fin
thf(fact_6706_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ~ ( member @ ( multiset @ A ) @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) ) ) ) ) ) ).
% subset_mset.Sup_fin.insert_not_elem
thf(fact_6707_subset__mset_OSup__fin_Oclosed,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X2: multiset @ A,Y2: multiset @ A] : ( member @ ( multiset @ A ) @ ( union_mset @ A @ X2 @ Y2 ) @ ( insert2 @ ( multiset @ A ) @ X2 @ ( insert2 @ ( multiset @ A ) @ Y2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ A4 ) ) ) ) ).
% subset_mset.Sup_fin.closed
thf(fact_6708_subset__mset_OSup__fin_Oinsert__remove,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.insert_remove
thf(fact_6709_subset__mset_OSup__fin_Oremove,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( member @ ( multiset @ A ) @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.remove
thf(fact_6710_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B3 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A5: multiset @ A,B4: multiset @ A] :
( ( Uu
= ( union_mset @ A @ A5 @ B4 ) )
& ( member @ ( multiset @ A ) @ A5 @ A4 )
& ( member @ ( multiset @ A ) @ B4 @ B3 ) ) ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
thf(fact_6711_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A5: multiset @ A] :
( ( Uu
= ( union_mset @ A @ X @ A5 ) )
& ( member @ ( multiset @ A ) @ A5 @ A4 ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf1_distrib
thf(fact_6712_subset__mset_OInf__fin_Osingleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.Inf_fin.singleton
thf(fact_6713_subset__mset_OInf__fin_Oinsert,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.Inf_fin.insert
thf(fact_6714_subset__mset_OInf__fin_Oeq__fold_H,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 )
= ( the2 @ ( multiset @ A )
@ ( finite_fold @ ( multiset @ A ) @ ( option @ ( multiset @ A ) )
@ ^ [X3: multiset @ A,Y3: option @ ( multiset @ A )] : ( some @ ( multiset @ A ) @ ( case_option @ ( multiset @ A ) @ ( multiset @ A ) @ X3 @ ( inter_mset @ A @ X3 ) @ Y3 ) )
@ ( none @ ( multiset @ A ) )
@ A4 ) ) ) ).
% subset_mset.Inf_fin.eq_fold'
thf(fact_6715_subset__mset_OInf__fin_Osubset,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B3 @ A4 )
=> ( ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B3 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset
thf(fact_6716_subset__mset_OInf__fin_Ohom__commute,axiom,
! [A: $tType,H3: ( multiset @ A ) > ( multiset @ A ),N4: set @ ( multiset @ A )] :
( ! [X2: multiset @ A,Y2: multiset @ A] :
( ( H3 @ ( inter_mset @ A @ X2 @ Y2 ) )
= ( inter_mset @ A @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H3 @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ N4 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H3 @ N4 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
thf(fact_6717_subset__mset_OInf__fin_Ounion,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) )
= ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B3 ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
thf(fact_6718_subset__mset_OcInf__eq__Inf__fin,axiom,
! [A: $tType,X7: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X7 )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ X7 ) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
thf(fact_6719_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ~ ( member @ ( multiset @ A ) @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_not_elem
thf(fact_6720_subset__mset_OInf__fin_Oclosed,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X2: multiset @ A,Y2: multiset @ A] : ( member @ ( multiset @ A ) @ ( inter_mset @ A @ X2 @ Y2 ) @ ( insert2 @ ( multiset @ A ) @ X2 @ ( insert2 @ ( multiset @ A ) @ Y2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) @ A4 ) ) ) ) ).
% subset_mset.Inf_fin.closed
thf(fact_6721_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A4 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_remove
thf(fact_6722_subset__mset_OInf__fin_Oremove,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( member @ ( multiset @ A ) @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A4 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.remove
thf(fact_6723_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) ) ) ) ).
% subset_mset.Inf_fin_le_Sup_fin
thf(fact_6724_subset__mset_OSup__fin_Osubset__imp,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B3 )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B3 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset_imp
thf(fact_6725_subset__mset_OcInf__eq__non__empty,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ X7 )
=> ( subseteq_mset @ A @ A3 @ X2 ) )
=> ( ! [Y2: multiset @ A] :
( ! [X5: multiset @ A] :
( ( member @ ( multiset @ A ) @ X5 @ X7 )
=> ( subseteq_mset @ A @ Y2 @ X5 ) )
=> ( subseteq_mset @ A @ Y2 @ A3 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X7 )
= A3 ) ) ) ) ).
% subset_mset.cInf_eq_non_empty
thf(fact_6726_subset__mset_OcInf__greatest,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ X7 )
=> ( subseteq_mset @ A @ Z2 @ X2 ) )
=> ( subseteq_mset @ A @ Z2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X7 ) ) ) ) ).
% subset_mset.cInf_greatest
thf(fact_6727_subset__mset_OcSup__least,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ X7 )
=> ( subseteq_mset @ A @ X2 @ Z2 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ X7 ) @ Z2 ) ) ) ).
% subset_mset.cSup_least
thf(fact_6728_subset__mset_OcSup__eq__non__empty,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ X7 )
=> ( subseteq_mset @ A @ X2 @ A3 ) )
=> ( ! [Y2: multiset @ A] :
( ! [X5: multiset @ A] :
( ( member @ ( multiset @ A ) @ X5 @ X7 )
=> ( subseteq_mset @ A @ X5 @ Y2 ) )
=> ( subseteq_mset @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X7 )
= A3 ) ) ) ) ).
% subset_mset.cSup_eq_non_empty
thf(fact_6729_subset__mset_OLeast__def,axiom,
! [A: $tType,P: ( multiset @ A ) > $o] :
( ( least @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ P )
= ( the @ ( multiset @ A )
@ ^ [X3: multiset @ A] :
( ( P @ X3 )
& ! [Y3: multiset @ A] :
( ( P @ Y3 )
=> ( subseteq_mset @ A @ X3 @ Y3 ) ) ) ) ) ).
% subset_mset.Least_def
thf(fact_6730_subset__mset_Ofinite__has__minimal,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ? [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ A4 )
& ! [Xa2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Xa2 @ A4 )
=> ( ( subseteq_mset @ A @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_6731_subset__mset_Ofinite__has__maximal,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ? [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ A4 )
& ! [Xa2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Xa2 @ A4 )
=> ( ( subseteq_mset @ A @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_6732_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A4: set @ B,M: multiset @ A,F2: B > ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subseteq_mset @ A @ M @ ( F2 @ X2 ) ) )
=> ( subseteq_mset @ A @ M @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_6733_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A4: set @ B,F2: B > ( multiset @ A ),M4: multiset @ A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subseteq_mset @ A @ ( F2 @ X2 ) @ M4 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ M4 ) ) ) ).
% subset_mset.cSUP_least
thf(fact_6734_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A4 )
=> ( subseteq_mset @ A @ X @ X3 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.bounded_iff
thf(fact_6735_subset__mset_OInf__fin_OboundedI,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [A8: multiset @ A] :
( ( member @ ( multiset @ A ) @ A8 @ A4 )
=> ( subseteq_mset @ A @ X @ A8 ) )
=> ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedI
thf(fact_6736_subset__mset_OInf__fin_OboundedE,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) )
=> ! [A15: multiset @ A] :
( ( member @ ( multiset @ A ) @ A15 @ A4 )
=> ( subseteq_mset @ A @ X @ A15 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
thf(fact_6737_subset__mset_OSup__fin_OboundedE,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ X )
=> ! [A15: multiset @ A] :
( ( member @ ( multiset @ A ) @ A15 @ A4 )
=> ( subseteq_mset @ A @ A15 @ X ) ) ) ) ) ).
% subset_mset.Sup_fin.boundedE
thf(fact_6738_subset__mset_OSup__fin_OboundedI,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [A8: multiset @ A] :
( ( member @ ( multiset @ A ) @ A8 @ A4 )
=> ( subseteq_mset @ A @ A8 @ X ) )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ X ) ) ) ) ).
% subset_mset.Sup_fin.boundedI
thf(fact_6739_subset__mset_OSup__fin_Obounded__iff,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A4 ) @ X )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A4 )
=> ( subseteq_mset @ A @ X3 @ X ) ) ) ) ) ) ).
% subset_mset.Sup_fin.bounded_iff
thf(fact_6740_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B3 )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B3 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
thf(fact_6741_Sup__multiset__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) )
= ( ^ [A6: set @ ( multiset @ A )] :
( if @ ( multiset @ A )
@ ( ( A6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A6 ) )
@ ( abs_multiset @ A
@ ^ [I3: A] :
( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X4: multiset @ A] : ( count @ A @ X4 @ I3 )
@ A6 ) ) )
@ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Sup_multiset_def
thf(fact_6742_Sup__multiset__in__multiset,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ord_less @ nat @ ( zero_zero @ nat )
@ ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [M9: multiset @ A] : ( count @ A @ M9 @ I3 )
@ A4 ) ) ) ) ) ) ) ).
% Sup_multiset_in_multiset
thf(fact_6743_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
thf(fact_6744_subset__mset_Obdd__above__UN,axiom,
! [A: $tType,B: $tType,I4: set @ B,A4: B > ( set @ ( multiset @ A ) )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( complete_Sup_Sup @ ( set @ ( multiset @ A ) ) @ ( image2 @ B @ ( set @ ( multiset @ A ) ) @ A4 @ I4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( A4 @ X3 ) ) ) ) ) ) ).
% subset_mset.bdd_above_UN
thf(fact_6745_subset__mset_Obdd__above__Un,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) )
= ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 ) ) ) ).
% subset_mset.bdd_above_Un
thf(fact_6746_subset__mset_Obdd__above__image__sup,axiom,
! [A: $tType,B: $tType,F2: B > ( multiset @ A ),G2: B > ( multiset @ A ),A4: set @ B] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X3: B] : ( union_mset @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 ) )
= ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A4 ) ) ) ) ).
% subset_mset.bdd_above_image_sup
thf(fact_6747_subset__mset_Obdd__above__Int2,axiom,
! [A: $tType,B3: set @ ( multiset @ A ),A4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) ) ) ).
% subset_mset.bdd_above_Int2
thf(fact_6748_subset__mset_Obdd__above__Int1,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) ) ) ).
% subset_mset.bdd_above_Int1
thf(fact_6749_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 )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ S ) @ A3 )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ S )
=> ( subseteq_mset @ A @ X3 @ A3 ) ) ) ) ) ) ).
% subset_mset.cSup_le_iff
thf(fact_6750_subset__mset_OcSup__mono,axiom,
! [A: $tType,B3: set @ ( multiset @ A ),A4: set @ ( multiset @ A )] :
( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ! [B7: multiset @ A] :
( ( member @ ( multiset @ A ) @ B7 @ B3 )
=> ? [X5: multiset @ A] :
( ( member @ ( multiset @ A ) @ X5 @ A4 )
& ( subseteq_mset @ A @ B7 @ X5 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ B3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.cSup_mono
thf(fact_6751_bdd__above__multiset__imp__bdd__above__count,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: A] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( condit941137186595557371_above @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X4: multiset @ A] : ( count @ A @ X4 @ X )
@ A4 ) ) ) ).
% bdd_above_multiset_imp_bdd_above_count
thf(fact_6752_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ B,G2: C > ( multiset @ A ),B3: set @ C,F2: B > ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G2 @ B3 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B3 )
& ( subseteq_mset @ A @ ( F2 @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G2 @ B3 ) ) ) ) ) ) ).
% subset_mset.cSUP_mono
thf(fact_6753_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),U: multiset @ A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ U )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( subseteq_mset @ A @ ( F2 @ X3 ) @ U ) ) ) ) ) ) ).
% subset_mset.cSUP_le_iff
thf(fact_6754_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) ) @ ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B3 ) ) ) ) ) ) ).
% subset_mset.cSup_inter_less_eq
thf(fact_6755_subset__mset_OcSup__subset__mono,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A4 @ B3 )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B3 ) ) ) ) ) ).
% subset_mset.cSup_subset_mono
thf(fact_6756_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 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ S )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X3: multiset @ A] :
! [Y3: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y3 @ S )
=> ( subseteq_mset @ A @ Y3 @ X3 ) ) ) ) ) ) ) ).
% subset_mset.cSup_cInf
thf(fact_6757_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A4: set @ B,G2: B > ( multiset @ A ),B3: set @ B,F2: B > ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subseteq_mset @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B3 ) ) ) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
thf(fact_6758_subset__mset_OcSup__insert__If,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),A3: multiset @ A] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X7 )
=> ( ( ( X7
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A3 @ X7 ) )
= A3 ) )
& ( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A3 @ X7 ) )
= ( union_mset @ A @ A3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X7 ) ) ) ) ) ) ).
% subset_mset.cSup_insert_If
thf(fact_6759_subset__mset_OcSup__insert,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X7 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A3 @ X7 ) )
= ( union_mset @ A @ A3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X7 ) ) ) ) ) ).
% subset_mset.cSup_insert
thf(fact_6760_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A4 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A4 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A5: B] : ( union_mset @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_6761_subset__mset_OcSup__union__distrib,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B3 ) ) ) ) ) ) ) ).
% subset_mset.cSup_union_distrib
thf(fact_6762_subset__mset_OcSUP__insert,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( union_mset @ A @ ( F2 @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) ) ) ) ) ).
% subset_mset.cSUP_insert
thf(fact_6763_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B3 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B3 ) ) ) ) ) ) ) ) ).
% subset_mset.cSUP_union
thf(fact_6764_set__mset__Sup,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( set_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ A4 ) ) ) ) ).
% set_mset_Sup
thf(fact_6765_count__Sup__multiset__nonempty,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),X: A] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( count @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) @ X )
= ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X4: multiset @ A] : ( count @ A @ X4 @ X )
@ A4 ) ) ) ) ) ).
% count_Sup_multiset_nonempty
thf(fact_6766_bdd__above__multiset__imp__finite__support,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( finite_finite2 @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( multiset @ A ) @ ( set @ A )
@ ^ [X4: multiset @ A] :
( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ X4 @ X3 ) ) )
@ A4 ) ) ) ) ) ).
% bdd_above_multiset_imp_finite_support
thf(fact_6767_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A4: C > ( multiset @ A ),I4: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A4 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F2 @ ( A4 @ X3 ) )
@ I4 ) )
@ ( F2 @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A4 @ I4 ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_6768_subset__mset_Omono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A4: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ ( multiset @ A ) @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) ) ) ) ) ) ) ).
% subset_mset.mono_cSup
thf(fact_6769_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_6770_subset__mset_Omono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ! [F2: ( multiset @ A ) > B,A4: multiset @ A,B3: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inter_mset @ A @ A4 @ B3 ) ) @ ( inf_inf @ B @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) ) ) ) ).
% subset_mset.mono_inf
thf(fact_6771_subset__mset_Omono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ! [F2: ( multiset @ A ) > B,A4: multiset @ A,B3: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( union_mset @ A @ A4 @ B3 ) ) ) ) ) ).
% subset_mset.mono_sup
thf(fact_6772_subset__mset_OGreatest__def,axiom,
! [A: $tType,P: ( multiset @ A ) > $o] :
( ( greatest @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ P )
= ( the @ ( multiset @ A )
@ ^ [X3: multiset @ A] :
( ( P @ X3 )
& ! [Y3: multiset @ A] :
( ( P @ Y3 )
=> ( subseteq_mset @ A @ Y3 @ X3 ) ) ) ) ) ).
% subset_mset.Greatest_def
thf(fact_6773_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A4: C > ( multiset @ A ),I4: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A4 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A4 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F2 @ ( A4 @ X3 ) )
@ I4 ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_6774_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
thf(fact_6775_subset__mset_Obdd__below__Un,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) )
= ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
& ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 ) ) ) ).
% subset_mset.bdd_below_Un
thf(fact_6776_subset__mset_Obdd__below__image__inf,axiom,
! [A: $tType,B: $tType,F2: B > ( multiset @ A ),G2: B > ( multiset @ A ),A4: set @ B] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X3: B] : ( inter_mset @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A4 ) )
= ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
& ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A4 ) ) ) ) ).
% subset_mset.bdd_below_image_inf
thf(fact_6777_subset__mset_Obdd__below__UN,axiom,
! [A: $tType,B: $tType,I4: set @ B,A4: B > ( set @ ( multiset @ A ) )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( complete_Sup_Sup @ ( set @ ( multiset @ A ) ) @ ( image2 @ B @ ( set @ ( multiset @ A ) ) @ A4 @ I4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( A4 @ X3 ) ) ) ) ) ) ).
% subset_mset.bdd_below_UN
thf(fact_6778_subset__mset_Obdd__below__Int1,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) ) ) ).
% subset_mset.bdd_below_Int1
thf(fact_6779_subset__mset_Obdd__below__Int2,axiom,
! [A: $tType,B3: set @ ( multiset @ A ),A4: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) ) ) ).
% subset_mset.bdd_below_Int2
thf(fact_6780_order_OGreatest_Ocong,axiom,
! [A: $tType] :
( ( greatest @ A )
= ( greatest @ A ) ) ).
% order.Greatest.cong
thf(fact_6781_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 )
=> ( ( subseteq_mset @ A @ A3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ S ) )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ S )
=> ( subseteq_mset @ A @ A3 @ X3 ) ) ) ) ) ) ).
% subset_mset.le_cInf_iff
thf(fact_6782_subset__mset_OcInf__mono,axiom,
! [A: $tType,B3: set @ ( multiset @ A ),A4: set @ ( multiset @ A )] :
( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ! [B7: multiset @ A] :
( ( member @ ( multiset @ A ) @ B7 @ B3 )
=> ? [X5: multiset @ A] :
( ( member @ ( multiset @ A ) @ X5 @ A4 )
& ( subseteq_mset @ A @ X5 @ B7 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B3 ) ) ) ) ) ).
% subset_mset.cInf_mono
thf(fact_6783_subset__mset_Ole__cINF__iff,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),U: multiset @ A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( subseteq_mset @ A @ U @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( subseteq_mset @ A @ U @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% subset_mset.le_cINF_iff
thf(fact_6784_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B3: set @ B,F2: C > ( multiset @ A ),A4: set @ C,G2: B > ( multiset @ A )] :
( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A4 )
& ( subseteq_mset @ A @ ( F2 @ X5 ) @ ( G2 @ M3 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B3 ) ) ) ) ) ) ).
% subset_mset.cINF_mono
thf(fact_6785_subset__mset_OcInf__superset__mono,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A4 @ B3 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ B3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.cInf_superset_mono
thf(fact_6786_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A4: set @ B,G2: B > ( multiset @ A ),B3: set @ B,F2: B > ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B3 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B3 )
=> ( subseteq_mset @ A @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) ) ) ) ) ) ).
% subset_mset.cINF_superset_mono
thf(fact_6787_subset__mset_OcInf__insert,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X7 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A3 @ X7 ) )
= ( inter_mset @ A @ A3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X7 ) ) ) ) ) ).
% subset_mset.cInf_insert
thf(fact_6788_subset__mset_OcInf__insert__If,axiom,
! [A: $tType,X7: set @ ( multiset @ A ),A3: multiset @ A] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X7 )
=> ( ( ( X7
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A3 @ X7 ) )
= A3 ) )
& ( ( X7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A3 @ X7 ) )
= ( inter_mset @ A @ A3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X7 ) ) ) ) ) ) ).
% subset_mset.cInf_insert_If
thf(fact_6789_subset__mset_OcInf__le__cSup,axiom,
! [A: $tType,A4: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A4 ) ) ) ) ) ).
% subset_mset.cInf_le_cSup
thf(fact_6790_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) ) ) ) ) ) ).
% subset_mset.less_eq_cInf_inter
thf(fact_6791_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A4 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A4 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A5: B] : ( inter_mset @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_6792_subset__mset_OcInf__union__distrib,axiom,
! [A: $tType,A4: set @ ( multiset @ A ),B3: set @ ( multiset @ A )] :
( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B3 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A4 @ B3 ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B3 ) ) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
thf(fact_6793_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 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ S )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X3: multiset @ A] :
! [Y3: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y3 @ S )
=> ( subseteq_mset @ A @ X3 @ Y3 ) ) ) ) ) ) ) ).
% subset_mset.cInf_cSup
thf(fact_6794_subset__mset_OcINF__insert,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inter_mset @ A @ ( F2 @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) ) ) ) ) ).
% subset_mset.cINF_insert
thf(fact_6795_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( B3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B3 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B3 ) ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B3 ) ) ) ) ) ) ) ) ).
% subset_mset.cINF_union
thf(fact_6796_subset__mset_Omono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A4: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ ( multiset @ A ) @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% subset_mset.mono_cInf
thf(fact_6797_subset__mset_Osum__nonneg__0,axiom,
! [B: $tType,A: $tType,S3: set @ B,F2: B > ( multiset @ A ),I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F2 @ S3 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( member @ B @ I @ S3 )
=> ( ( F2 @ I )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_0
thf(fact_6798_subset__mset_Osum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType,S3: set @ B,F2: B > ( multiset @ A ),B3: multiset @ A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F2 @ S3 )
= B3 )
=> ( ( member @ B @ I @ S3 )
=> ( subseteq_mset @ A @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% subset_mset.sum_nonneg_leq_bound
thf(fact_6799_subset__mset_Osum__mono,axiom,
! [A: $tType,B: $tType,K5: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ! [I2: B] :
( ( member @ B @ I2 @ K5 )
=> ( subseteq_mset @ A @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F2 @ K5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G2 @ K5 ) ) ) ).
% subset_mset.sum_mono
thf(fact_6800_subset__mset_OatLeast__eq__UNIV__iff,axiom,
! [A: $tType,X: multiset @ A] :
( ( ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X )
= ( top_top @ ( set @ ( multiset @ A ) ) ) )
= ( X
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.atLeast_eq_UNIV_iff
thf(fact_6801_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subset_mset @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F2 @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G2 @ A4 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_6802_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( lattic4895041142388067077er_set @ ( multiset @ A ) @ ( union_mset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.Sup_fin.semilattice_order_set_axioms
thf(fact_6803_subset__mset_OatLeast__def,axiom,
! [A: $tType,L: multiset @ A] :
( ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L )
= ( collect @ ( multiset @ A ) @ ( subseteq_mset @ A @ L ) ) ) ).
% subset_mset.atLeast_def
thf(fact_6804_subset__mset_Oordering__top__axioms,axiom,
! [A: $tType] :
( ordering_top @ ( multiset @ A )
@ ^ [A6: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ B5 @ A6 )
@ ^ [A6: multiset @ A,B5: multiset @ A] : ( subset_mset @ A @ B5 @ A6 )
@ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.ordering_top_axioms
thf(fact_6805_wf__subset__mset__rel,axiom,
! [A: $tType] : ( wf @ ( multiset @ A ) @ ( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o @ ( subset_mset @ A ) ) ) ) ).
% wf_subset_mset_rel
thf(fact_6806_subset__mset_Oasymp__greater,axiom,
! [A: $tType] :
( asymp @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.asymp_greater
thf(fact_6807_ord_OatLeast__def,axiom,
! [A: $tType] :
( ( set_atLeast @ A )
= ( ^ [Less_eq2: A > A > $o,L2: A] : ( collect @ A @ ( Less_eq2 @ L2 ) ) ) ) ).
% ord.atLeast_def
thf(fact_6808_subset__implies__mult,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( subset_mset @ A @ A4 @ B3 )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A4 @ B3 ) @ ( mult @ A @ R3 ) ) ) ).
% subset_implies_mult
thf(fact_6809_subset__mset_OacyclicI__order,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ B ),F2: B > ( multiset @ A )] :
( ! [A8: B,B7: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A8 @ B7 ) @ R3 )
=> ( subset_mset @ A @ ( F2 @ B7 ) @ ( F2 @ A8 ) ) )
=> ( transitive_acyclic @ B @ R3 ) ) ).
% subset_mset.acyclicI_order
thf(fact_6810_subset__mset_Olexordp_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o ) @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o )
@ ^ [P6: ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o,X12: list @ ( multiset @ A ),X23: list @ ( multiset @ A )] :
( ? [Y3: multiset @ A,Ys2: list @ ( multiset @ A )] :
( ( X12
= ( nil @ ( multiset @ A ) ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y3 @ Ys2 ) ) )
| ? [X3: multiset @ A,Y3: multiset @ A,Xs2: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X3 @ Xs2 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y3 @ Ys2 ) )
& ( subset_mset @ A @ X3 @ Y3 ) )
| ? [X3: multiset @ A,Y3: multiset @ A,Xs2: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X3 @ Xs2 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y3 @ Ys2 ) )
& ~ ( subset_mset @ A @ X3 @ Y3 )
& ~ ( subset_mset @ A @ Y3 @ X3 )
& ( P6 @ Xs2 @ Ys2 ) ) ) ) ).
% subset_mset.lexordp.mono
thf(fact_6811_subset__mset_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ ( multiset @ A ) @ ( subset_mset @ A ) )
= ( complete_lattice_lfp @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o )
@ ^ [P6: ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o,X12: list @ ( multiset @ A ),X23: list @ ( multiset @ A )] :
( ? [Y3: multiset @ A,Ys2: list @ ( multiset @ A )] :
( ( X12
= ( nil @ ( multiset @ A ) ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y3 @ Ys2 ) ) )
| ? [X3: multiset @ A,Y3: multiset @ A,Xs2: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X3 @ Xs2 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y3 @ Ys2 ) )
& ( subset_mset @ A @ X3 @ Y3 ) )
| ? [X3: multiset @ A,Y3: multiset @ A,Xs2: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X3 @ Xs2 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y3 @ Ys2 ) )
& ~ ( subset_mset @ A @ X3 @ Y3 )
& ~ ( subset_mset @ A @ Y3 @ X3 )
& ( P6 @ Xs2 @ Ys2 ) ) ) ) ) ).
% subset_mset.lexordp_def
thf(fact_6812_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
thf(fact_6813_subset__mset_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( semila1105856199041335345_order @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) )
@ ^ [A6: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ B5 @ A6 )
@ ^ [A6: multiset @ A,B5: multiset @ A] : ( subset_mset @ A @ B5 @ A6 ) ) ).
% subset_mset.semilattice_neutr_order_axioms
thf(fact_6814_subset__mset_OcSUP__lessD,axiom,
! [B: $tType,A: $tType,F2: B > ( multiset @ A ),A4: set @ B,Y: multiset @ A,I: B] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( subset_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( subset_mset @ A @ ( F2 @ I ) @ Y ) ) ) ) ).
% subset_mset.cSUP_lessD
thf(fact_6815_subset__mset_Oless__cINF__D,axiom,
! [A: $tType,B: $tType,F2: B > ( multiset @ A ),A4: set @ B,Y: multiset @ A,I: B] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) )
=> ( ( subset_mset @ A @ Y @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( subset_mset @ A @ Y @ ( F2 @ I ) ) ) ) ) ).
% subset_mset.less_cINF_D
thf(fact_6816_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I4: set @ B,F2: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F2 @ I2 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F2 @ I4 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_6817_subset__mset_OgreaterThanLessThan__empty,axiom,
! [A: $tType,L: multiset @ A,K: multiset @ A] :
( ( subseteq_mset @ A @ L @ K )
=> ( ( set_gr287244882034783167ssThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ K @ L )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.greaterThanLessThan_empty
thf(fact_6818_subset__mset_OIio__Int__singleton,axiom,
! [A: $tType,X: multiset @ A,K: multiset @ A] :
( ( ( subset_mset @ A @ X @ K )
=> ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ K ) @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
& ( ~ ( subset_mset @ A @ X @ K )
=> ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ K ) @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.Iio_Int_singleton
thf(fact_6819_subset__mset_OlessThan__def,axiom,
! [A: $tType,U: multiset @ A] :
( ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ U )
= ( collect @ ( multiset @ A )
@ ^ [X3: multiset @ A] : ( subset_mset @ A @ X3 @ U ) ) ) ).
% subset_mset.lessThan_def
thf(fact_6820_ord_OlessThan__def,axiom,
! [A: $tType] :
( ( set_lessThan @ A )
= ( ^ [Less2: A > A > $o,U2: A] :
( collect @ A
@ ^ [X3: A] : ( Less2 @ X3 @ U2 ) ) ) ) ).
% ord.lessThan_def
thf(fact_6821_ord_OatLeastLessThan__def,axiom,
! [A: $tType] :
( ( set_atLeastLessThan @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq2 @ L2 ) @ ( set_lessThan @ A @ Less2 @ U2 ) ) ) ) ).
% ord.atLeastLessThan_def
thf(fact_6822_subset__mset_OatLeastLessThan__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ U ) ) ) ).
% subset_mset.atLeastLessThan_def
thf(fact_6823_subset__mset_OatLeastLessThan__empty,axiom,
! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
( ( 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
thf(fact_6824_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 ) ) ) )
= ( ~ ( subset_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastLessThan_empty_iff
thf(fact_6825_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 ) )
= ( ~ ( subset_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastLessThan_empty_iff2
thf(fact_6826_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 )
= ( minus_minus @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( insert2 @ ( multiset @ A ) @ B2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_6827_subset__mset_OgreaterThanLessThan__eq,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( set_gr287244882034783167ssThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ A3 ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ B2 ) ) ) ).
% subset_mset.greaterThanLessThan_eq
thf(fact_6828_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 ) ) ) )
= ( ~ ( subseteq_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastatMost_empty_iff
thf(fact_6829_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 ) )
= ( ~ ( subseteq_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastatMost_empty_iff2
thf(fact_6830_subset__mset_OatLeastatMost__empty,axiom,
! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
( ( subset_mset @ A @ B2 @ A3 )
=> ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastatMost_empty
thf(fact_6831_subset__mset_OatLeastAtMost__singleton,axiom,
! [A: $tType,A3: multiset @ A] :
( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ A3 )
= ( insert2 @ ( multiset @ A ) @ A3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastAtMost_singleton
thf(fact_6832_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 )
= ( insert2 @ ( multiset @ A ) @ C2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ).
% subset_mset.atLeastAtMost_singleton_iff
thf(fact_6833_subset__mset_OgreaterThan__def,axiom,
! [A: $tType,L: multiset @ A] :
( ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L )
= ( collect @ ( multiset @ A ) @ ( subset_mset @ A @ L ) ) ) ).
% subset_mset.greaterThan_def
thf(fact_6834_ord_OgreaterThan__def,axiom,
! [A: $tType] :
( ( set_greaterThan @ A )
= ( ^ [Less2: A > A > $o,L2: A] : ( collect @ A @ ( Less2 @ L2 ) ) ) ) ).
% ord.greaterThan_def
thf(fact_6835_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 )
= ( insert2 @ ( multiset @ A ) @ A3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.atLeastAtMost_singleton'
thf(fact_6836_ord_OgreaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( set_gr287244882034783167ssThan @ A )
= ( ^ [Less2: A > A > $o,A5: A,B4: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ A5 ) @ ( set_lessThan @ A @ Less2 @ B4 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_6837_ord_OgreaterThanLessThan__def,axiom,
! [A: $tType] :
( ( set_gr287244882034783167ssThan @ A )
= ( ^ [Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ L2 ) @ ( set_lessThan @ A @ Less2 @ U2 ) ) ) ) ).
% ord.greaterThanLessThan_def
thf(fact_6838_subset__mset_OgreaterThanLessThan__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_gr287244882034783167ssThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ U ) ) ) ).
% subset_mset.greaterThanLessThan_def
thf(fact_6839_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 )
= ( minus_minus @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( insert2 @ ( multiset @ A ) @ A3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_6840_ord_OatLeastAtMost__def,axiom,
! [A: $tType] :
( ( set_atLeastAtMost @ A )
= ( ^ [Less_eq2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq2 @ L2 ) @ ( set_atMost @ A @ Less_eq2 @ U2 ) ) ) ) ).
% ord.atLeastAtMost_def
thf(fact_6841_subset__mset_OgreaterThanAtMost__empty,axiom,
! [A: $tType,L: multiset @ A,K: multiset @ A] :
( ( 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
thf(fact_6842_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 ) ) ) )
= ( ~ ( subset_mset @ A @ K @ L ) ) ) ).
% subset_mset.greaterThanAtMost_empty_iff
thf(fact_6843_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 ) )
= ( ~ ( subset_mset @ A @ K @ L ) ) ) ).
% subset_mset.greaterThanAtMost_empty_iff2
thf(fact_6844_subset__mset_OatMost__def,axiom,
! [A: $tType,U: multiset @ A] :
( ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ U )
= ( collect @ ( multiset @ A )
@ ^ [X3: multiset @ A] : ( subseteq_mset @ A @ X3 @ U ) ) ) ).
% subset_mset.atMost_def
thf(fact_6845_ord_OatMost__def,axiom,
! [A: $tType] :
( ( set_atMost @ A )
= ( ^ [Less_eq2: A > A > $o,U2: A] :
( collect @ A
@ ^ [X3: A] : ( Less_eq2 @ X3 @ U2 ) ) ) ) ).
% ord.atMost_def
thf(fact_6846_ord_OgreaterThanAtMost__def,axiom,
! [A: $tType] :
( ( set_gr3752724095348155675AtMost @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ L2 ) @ ( set_atMost @ A @ Less_eq2 @ U2 ) ) ) ) ).
% ord.greaterThanAtMost_def
thf(fact_6847_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
! [A: $tType,H3: multiset @ A] :
( ( bot_bot @ ( set @ ( multiset @ A ) ) )
!= ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ H3 ) ) ).
% subset_mset.not_empty_eq_Iic_eq_empty
thf(fact_6848_subset__mset_OgreaterThanAtMost__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L ) @ ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ U ) ) ) ).
% subset_mset.greaterThanAtMost_def
thf(fact_6849_subset__mset_OatLeastAtMost__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L ) @ ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ U ) ) ) ).
% subset_mset.atLeastAtMost_def
thf(fact_6850_folding__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding @ A @ B )
= ( finite_folding_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% folding_def'
thf(fact_6851_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F: A > ( product_prod @ B @ C ),G: B > C > D,X3: A] : ( G @ ( product_fst @ B @ C @ ( F @ X3 ) ) @ ( product_snd @ B @ C @ ( F @ X3 ) ) ) ) ) ).
% scomp_unfold
thf(fact_6852_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F2: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F2 )
= ( F2 @ X ) ) ).
% Pair_scomp
thf(fact_6853_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_6854_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F4: $tType,E: $tType,F2: A > ( product_prod @ E @ F4 ),G2: E > F4 > ( product_prod @ C @ D ),H3: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F4 @ ( product_prod @ C @ D ) @ F2 @ G2 ) @ H3 )
= ( product_scomp @ A @ E @ F4 @ B @ F2
@ ^ [X3: E] : ( product_scomp @ F4 @ C @ D @ B @ ( G2 @ X3 ) @ H3 ) ) ) ).
% scomp_scomp
thf(fact_6855_card_Ofolding__axioms,axiom,
! [A: $tType] :
( finite_folding @ A @ nat
@ ^ [Uu: A] : suc ) ).
% card.folding_axioms
thf(fact_6856_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,B3: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F2 @ B3 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_6857_cr__int__def,axiom,
( cr_int
= ( ^ [X3: product_prod @ nat @ nat] :
( ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( abs_Integ @ X3 ) ) ) ) ).
% cr_int_def
thf(fact_6858_Inf__fin__def,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% Inf_fin_def
thf(fact_6859_Sup__fin__def,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% Sup_fin_def
thf(fact_6860_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( Less_eq @ X @ X3 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_6861_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( Less_eq @ X @ A8 ) )
=> ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_6862_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A4 )
=> ( Less_eq @ X @ A15 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_6863_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( F2 @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A4 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_6864_semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_6865_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% Sup_fin.semilattice_set_axioms
thf(fact_6866_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% Inf_fin.semilattice_set_axioms
thf(fact_6867_semilattice__set_Osingleton,axiom,
! [A: $tType,F2: A > A > A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_6868_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F2: A > A > A,H3: A > A,N4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ! [X2: A,Y2: A] :
( ( H3 @ ( F2 @ X2 @ Y2 ) )
= ( F2 @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ N4 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ ( image2 @ A @ A @ H3 @ N4 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_6869_semilattice__set_Osubset,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,B3: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ B3 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_6870_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_6871_semilattice__set_Oinsert,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_6872_semilattice__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member @ A @ ( F2 @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_6873_semilattice__set_Ounion,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,B3: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_6874_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_6875_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_6876_antisymp__antisym__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
= ( antisym @ A @ R3 ) ) ).
% antisymp_antisym_eq
thf(fact_6877_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_6878_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_6879_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_6880_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_6881_antisym__bot,axiom,
! [A: $tType] : ( antisymp @ A @ ( bot_bot @ ( A > A > $o ) ) ) ).
% antisym_bot
thf(fact_6882_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_6883_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_6884_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6885_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) ) ) ) ).
% insort_insert_insort
thf(fact_6886_in__range_Osimps,axiom,
! [H3: heap_ext @ product_unit,As: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
= ( ! [X3: nat] :
( ( member @ nat @ X3 @ As )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% in_range.simps
thf(fact_6887_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs )
= Xs ) ) ) ).
% insort_insert_triv
thf(fact_6888_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_6889_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_6890_in__range_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ! [X2: nat] :
( ( member @ nat @ X2 @ As4 )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_6891_in__range_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ~ ! [X5: nat] :
( ( member @ nat @ X5 @ As4 )
=> ( ord_less @ nat @ X5 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_6892_in__range_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( Y
= ( ~ ! [X3: nat] :
( ( member @ nat @ X3 @ As4 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_6893_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A4: set @ B,F2: B > A] :
( ( bind3 @ B @ A @ A4
@ ^ [X3: B] : ( insert2 @ A @ ( F2 @ X3 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F2 @ A4 ) ) ).
% bind_singleton_conv_image
thf(fact_6894_surj__prod__encode,axiom,
( ( image2 @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_prod_encode
thf(fact_6895_empty__bind,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A )] :
( ( bind3 @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_6896_Set_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B3: A > ( set @ C ),C3: C > ( set @ B )] :
( ( bind3 @ C @ B @ ( bind3 @ A @ C @ A4 @ B3 ) @ C3 )
= ( bind3 @ A @ B @ A4
@ ^ [X3: A] : ( bind3 @ C @ B @ ( B3 @ X3 ) @ C3 ) ) ) ).
% Set.bind_bind
thf(fact_6897_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind3 @ A @ B @ A4
@ ^ [Uu: A] : B3 )
= B3 ) ) ).
% nonempty_bind_const
thf(fact_6898_bind__const,axiom,
! [B: $tType,A: $tType,A4: set @ B,B3: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind3 @ B @ A @ A4
@ ^ [Uu: B] : B3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind3 @ B @ A @ A4
@ ^ [Uu: B] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_6899_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_6900_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_6901_Set_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind3 @ A @ B )
= ( ^ [A6: set @ A,F: A > ( set @ B )] :
( collect @ B
@ ^ [X3: B] :
? [Y3: set @ B] :
( ( member @ ( set @ B ) @ Y3 @ ( image2 @ A @ ( set @ B ) @ F @ A6 ) )
& ( member @ B @ X3 @ Y3 ) ) ) ) ) ).
% Set.bind_def
thf(fact_6902_bij__prod__encode,axiom,
bij_betw @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_prod_encode
thf(fact_6903_list__encode_Oelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X2: nat,Xs3: list @ nat] :
( ( X
= ( cons @ nat @ X2 @ Xs3 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_6904_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M2: nat,N2: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N2 ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_6905_surj__list__encode,axiom,
( ( image2 @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_list_encode
thf(fact_6906_bij__list__encode,axiom,
bij_betw @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_list_encode
thf(fact_6907_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_6908_list__encode_Opelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
=> ( ( ( X
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X2: nat,Xs3: list @ nat] :
( ( X
= ( cons @ nat @ X2 @ Xs3 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X2 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_6909_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X3: A] : X3 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6910_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A4 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_6911_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A4 )
= ( nil @ B ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_6912_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F2: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_6913_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( insert2 @ B @ X @ A4 ) )
= ( insort_key @ A @ B @ Less_eq @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_6914_String_Oless__literal__def,axiom,
( ( ord_less @ literal )
= ( map_fun @ literal @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ explode @ ( map_fun @ literal @ ( list @ char ) @ $o @ $o @ explode @ ( id @ $o ) )
@ ( lexordp2 @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) ) ) ) ).
% String.less_literal_def
thf(fact_6915_less__literal_Orep__eq,axiom,
( ( ord_less @ literal )
= ( ^ [X3: literal,Xa4: literal] :
( lexordp2 @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) )
@ ( explode @ X3 )
@ ( explode @ Xa4 ) ) ) ) ).
% less_literal.rep_eq
thf(fact_6916_String_Oless__eq__literal__def,axiom,
( ( ord_less_eq @ literal )
= ( map_fun @ literal @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ explode @ ( map_fun @ literal @ ( list @ char ) @ $o @ $o @ explode @ ( id @ $o ) )
@ ( lexordp_eq @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) ) ) ) ).
% String.less_eq_literal_def
thf(fact_6917_less__literal_Otransfer,axiom,
( bNF_rel_fun @ ( list @ char ) @ literal @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ pcr_literal
@ ( bNF_rel_fun @ ( list @ char ) @ literal @ $o @ $o @ pcr_literal
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( lexordp2 @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) )
@ ( ord_less @ literal ) ) ).
% less_literal.transfer
thf(fact_6918_literal_Orep__transfer,axiom,
( bNF_rel_fun @ ( list @ char ) @ literal @ ( list @ char ) @ ( list @ char ) @ pcr_literal
@ ( list_all2 @ char @ char
@ ^ [Y5: char,Z4: char] : Y5 = Z4 )
@ ^ [X3: list @ char] : X3
@ explode ) ).
% literal.rep_transfer
thf(fact_6919_less__eq__literal_Otransfer,axiom,
( bNF_rel_fun @ ( list @ char ) @ literal @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ pcr_literal
@ ( bNF_rel_fun @ ( list @ char ) @ literal @ $o @ $o @ pcr_literal
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( lexordp_eq @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) )
@ ( ord_less_eq @ literal ) ) ).
% less_eq_literal.transfer
thf(fact_6920_less__eq__literal_Orep__eq,axiom,
( ( ord_less_eq @ literal )
= ( ^ [X3: literal,Xa4: literal] :
( lexordp_eq @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) )
@ ( explode @ X3 )
@ ( explode @ Xa4 ) ) ) ) ).
% less_eq_literal.rep_eq
thf(fact_6921_String_Ocr__literal__def,axiom,
( cr_literal
= ( ^ [X3: list @ char,Y3: literal] :
( X3
= ( explode @ Y3 ) ) ) ) ).
% String.cr_literal_def
thf(fact_6922_MOST__eq_I2_J,axiom,
! [A: $tType,A3: A] :
( ( eventually @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ A3 )
@ ( cofinite @ A ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% MOST_eq(2)
thf(fact_6923_cofinite__bot,axiom,
! [A: $tType] :
( ( ( cofinite @ A )
= ( bot_bot @ ( filter @ A ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% cofinite_bot
thf(fact_6924_MOST__const,axiom,
! [A: $tType,P: $o] :
( ( eventually @ A
@ ^ [X3: A] : P
@ ( cofinite @ A ) )
= ( P
| ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% MOST_const
thf(fact_6925_MOST__eq_I1_J,axiom,
! [A: $tType,A3: A] :
( ( eventually @ A
@ ^ [X3: A] : X3 = A3
@ ( cofinite @ A ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% MOST_eq(1)
thf(fact_6926_MOST__conj__distrib,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) )
@ ( cofinite @ A ) )
= ( ( eventually @ A @ P @ ( cofinite @ A ) )
& ( eventually @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% MOST_conj_distrib
thf(fact_6927_MOST__imp__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ ( cofinite @ A ) )
= ( eventually @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% MOST_imp_iff
thf(fact_6928_MOST__rev__mp,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ ( cofinite @ A ) )
=> ( eventually @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% MOST_rev_mp
thf(fact_6929_MOST__conjI,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A @ Q2 @ ( cofinite @ A ) )
=> ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) )
@ ( cofinite @ A ) ) ) ) ).
% MOST_conjI
thf(fact_6930_MOST__mono,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( eventually @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% MOST_mono
thf(fact_6931_ALL__MOST,axiom,
! [A: $tType,P: A > $o] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ ( cofinite @ A ) ) ) ).
% ALL_MOST
thf(fact_6932_MOST__I,axiom,
! [A: $tType,P: A > $o] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ ( cofinite @ A ) ) ) ).
% MOST_I
thf(fact_6933_MOST__eq__imp_I1_J,axiom,
! [A: $tType,A3: A,P: A > $o] :
( eventually @ A
@ ^ [X3: A] :
( ( X3 = A3 )
=> ( P @ X3 ) )
@ ( cofinite @ A ) ) ).
% MOST_eq_imp(1)
thf(fact_6934_MOST__eq__imp_I2_J,axiom,
! [A: $tType,A3: A,P: A > $o] :
( eventually @ A
@ ^ [X3: A] :
( ( A3 = X3 )
=> ( P @ X3 ) )
@ ( cofinite @ A ) ) ).
% MOST_eq_imp(2)
thf(fact_6935_MOST__neq_I1_J,axiom,
! [A: $tType,A3: A] :
( eventually @ A
@ ^ [X3: A] : X3 != A3
@ ( cofinite @ A ) ) ).
% MOST_neq(1)
thf(fact_6936_MOST__neq_I2_J,axiom,
! [A: $tType,A3: A] :
( eventually @ A
@ ^ [X3: A] : A3 != X3
@ ( cofinite @ A ) ) ).
% MOST_neq(2)
thf(fact_6937_MOST__iff__finiteNeg,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) ) ).
% MOST_iff_finiteNeg
thf(fact_6938_eventually__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) ) ).
% eventually_cofinite
thf(fact_6939_MOST__SucD,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ ( cofinite @ nat ) )
=> ( eventually @ nat @ P @ ( cofinite @ nat ) ) ) ).
% MOST_SucD
thf(fact_6940_MOST__SucI,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ ( cofinite @ nat ) ) ) ).
% MOST_SucI
thf(fact_6941_MOST__Suc__iff,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ ( cofinite @ nat ) )
= ( eventually @ nat @ P @ ( cofinite @ nat ) ) ) ).
% MOST_Suc_iff
thf(fact_6942_MOST__nat,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
= ( ? [M2: nat] :
! [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat
thf(fact_6943_MOST__ge__nat,axiom,
! [M: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ M ) @ ( cofinite @ nat ) ) ).
% MOST_ge_nat
thf(fact_6944_MOST__nat__le,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
= ( ? [M2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat_le
thf(fact_6945_cofinite__def,axiom,
! [A: $tType] :
( ( cofinite @ A )
= ( abs_filter @ A
@ ^ [P2: A > $o] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P2 @ X3 ) ) ) ) ) ).
% cofinite_def
thf(fact_6946_MOST__finite__Ball__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B
@ ^ [Y3: B] :
! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 @ Y3 ) )
@ ( cofinite @ B ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( eventually @ B @ ( P @ X3 ) @ ( cofinite @ B ) ) ) ) ) ) ).
% MOST_finite_Ball_distrib
thf(fact_6947_MOST__inj,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( eventually @ B
@ ^ [X3: B] : ( P @ ( F2 @ X3 ) )
@ ( cofinite @ B ) ) ) ) ).
% MOST_inj
thf(fact_6948_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_6949_less__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( ( list @ char ) > $o )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( lexordp2 @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) )
@ ( lexordp2 @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) ) ) ).
% less_literal.rsp
thf(fact_6950_plus__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( ( list @ char ) > ( list @ char ) ) @ ( ( list @ char ) > ( list @ char ) )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ char ) @ ( list @ char )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
@ ( append @ char )
@ ( append @ char ) ) ).
% plus_literal.rsp
thf(fact_6951_asciis__of__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ code_integer ) @ ( list @ code_integer )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y5: list @ code_integer,Z4: list @ code_integer] : Y5 = Z4
@ ( map @ char @ code_integer @ ( comm_s6883823935334413003f_char @ code_integer ) )
@ ( map @ char @ code_integer @ ( comm_s6883823935334413003f_char @ code_integer ) ) ) ).
% asciis_of_literal.rsp
thf(fact_6952_size__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ nat @ nat
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( size_size @ ( list @ char ) )
@ ( size_size @ ( list @ char ) ) ) ).
% size_literal.rsp
thf(fact_6953_Literal_Orsp,axiom,
( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > ( list @ char ) > ( list @ char ) ) @ ( $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( ( list @ char ) > ( list @ char ) ) @ ( ( list @ char ) > ( list @ char ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ char ) @ ( list @ char )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) ) ) ).
% Literal.rsp
thf(fact_6954_literal_Oexplode,axiom,
! [X: literal] :
( member @ ( list @ char ) @ ( explode @ X )
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ).
% literal.explode
thf(fact_6955_literal_Oexplode__cases,axiom,
! [Y: list @ char] :
( ( member @ ( list @ char ) @ Y
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ~ ! [X2: literal] :
( Y
!= ( explode @ X2 ) ) ) ).
% literal.explode_cases
thf(fact_6956_literal_Oexplode__induct,axiom,
! [Y: list @ char,P: ( list @ char ) > $o] :
( ( member @ ( list @ char ) @ Y
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ! [X2: literal] : ( P @ ( explode @ X2 ) )
=> ( P @ Y ) ) ) ).
% literal.explode_induct
thf(fact_6957_group_Oleft__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A,C2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A3 @ B2 )
= ( F2 @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% group.left_cancel
thf(fact_6958_group_Oleft__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ ( Inverse @ A3 ) @ A3 )
= Z2 ) ) ).
% group.left_inverse
thf(fact_6959_group_Oright__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,B2: A,A3: A,C2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ B2 @ A3 )
= ( F2 @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% group.right_cancel
thf(fact_6960_group_Oright__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ A3 @ ( Inverse @ A3 ) )
= Z2 ) ) ).
% group.right_inverse
thf(fact_6961_group_Oinverse__unique,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A3 @ B2 )
= Z2 )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_6962_group_Oinverse__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_6963_group_Oinverse__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ Z2 )
= Z2 ) ) ).
% group.inverse_neutral
thf(fact_6964_group_Ogroup__left__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ Z2 @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_6965_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ ( F2 @ A3 @ B2 ) )
= ( F2 @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_6966_zero__literal_Orsp,axiom,
( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ ( nil @ char )
@ ( nil @ char ) ) ).
% zero_literal.rsp
thf(fact_6967_literal_Odomain__eq,axiom,
( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= ( ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ).
% literal.domain_eq
thf(fact_6968_literal_Odomain,axiom,
( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= ( ^ [X3: list @ char] :
? [Y3: list @ char] :
( ( list_all2 @ char @ char
@ ^ [Y5: char,Z4: char] : Y5 = Z4
@ X3
@ Y3 )
& ! [Z5: char] :
( ( member @ char @ Z5 @ ( set2 @ char @ Y3 ) )
=> ~ ( digit7 @ Z5 ) ) ) ) ) ).
% literal.domain
thf(fact_6969_less__eq__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( ( list @ char ) > $o )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( lexordp_eq @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) )
@ ( lexordp_eq @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) ) ) ) ).
% less_eq_literal.rsp
thf(fact_6970_less__eq__literal_Oabs__eq,axiom,
! [Xa: list @ char,X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( ord_less_eq @ literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( lexordp_eq @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) )
@ Xa
@ X ) ) ) ) ).
% less_eq_literal.abs_eq
thf(fact_6971_less__literal_Oabs__eq,axiom,
! [Xa: list @ char,X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( ord_less @ literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( lexordp2 @ char
@ ^ [C5: char,D5: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C5 ) @ ( comm_s6883823935334413003f_char @ nat @ D5 ) )
@ Xa
@ X ) ) ) ) ).
% less_literal.abs_eq
thf(fact_6972_literal_OAbs__literal__cases,axiom,
! [X: literal] :
~ ! [Y2: list @ char] :
( ( X
= ( abs_literal @ Y2 ) )
=> ~ ( member @ ( list @ char ) @ Y2
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ) ).
% literal.Abs_literal_cases
thf(fact_6973_literal_OAbs__literal__induct,axiom,
! [P: literal > $o,X: literal] :
( ! [Y2: list @ char] :
( ( member @ ( list @ char ) @ Y2
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( P @ ( abs_literal @ Y2 ) ) )
=> ( P @ X ) ) ).
% literal.Abs_literal_induct
thf(fact_6974_literal_OAbs__literal__inject,axiom,
! [X: list @ char,Y: list @ char] :
( ( member @ ( list @ char ) @ X
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ( member @ ( list @ char ) @ Y
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ( ( abs_literal @ X )
= ( abs_literal @ Y ) )
= ( X = Y ) ) ) ) ).
% literal.Abs_literal_inject
thf(fact_6975_literal_Otype__definition__literal,axiom,
( type_definition @ literal @ ( list @ char ) @ explode @ abs_literal
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ).
% literal.type_definition_literal
thf(fact_6976_literal_OAbs__literal__inverse,axiom,
! [Y: list @ char] :
( ( member @ ( list @ char ) @ Y
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ( explode @ ( abs_literal @ Y ) )
= Y ) ) ).
% literal.Abs_literal_inverse
thf(fact_6977_plus__literal_Oabs__eq,axiom,
! [Xa: list @ char,X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( plus_plus @ literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( abs_literal @ ( append @ char @ Xa @ X ) ) ) ) ) ).
% plus_literal.abs_eq
thf(fact_6978_String_OQuotient__literal,axiom,
( quotient @ ( list @ char ) @ literal
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ abs_literal
@ explode
@ cr_literal ) ).
% String.Quotient_literal
thf(fact_6979_asciis__of__literal_Oabs__eq,axiom,
! [X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( asciis_of_literal @ ( abs_literal @ X ) )
= ( map @ char @ code_integer @ ( comm_s6883823935334413003f_char @ code_integer ) @ X ) ) ) ).
% asciis_of_literal.abs_eq
thf(fact_6980_size__literal_Oabs__eq,axiom,
! [X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( size_size @ literal @ ( abs_literal @ X ) )
= ( size_size @ ( list @ char ) @ X ) ) ) ).
% size_literal.abs_eq
thf(fact_6981_literal_Odomain__par__left__total,axiom,
! [P7: ( list @ char ) > $o] :
( ( left_total @ ( list @ char ) @ ( list @ char )
@ ( list_all2 @ char @ char
@ ^ [Y5: char,Z4: char] : Y5 = Z4 ) )
=> ( ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( list_all2 @ char @ char
@ ^ [Y5: char,Z4: char] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P7
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
=> ( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= P7 ) ) ) ).
% literal.domain_par_left_total
thf(fact_6982_literal_Odomain__par,axiom,
! [DR: char > $o,P24: ( list @ char ) > $o] :
( ( ( domainp @ char @ char
@ ^ [Y5: char,Z4: char] : Y5 = Z4 )
= DR )
=> ( ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( list_all2 @ char @ char
@ ^ [Y5: char,Z4: char] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P24
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
=> ( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= ( inf_inf @ ( ( list @ char ) > $o ) @ ( list_all @ char @ DR ) @ P24 ) ) ) ) ).
% literal.domain_par
thf(fact_6983_list_Opred__True,axiom,
! [A: $tType] :
( ( list_all @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: list @ A] : $true ) ) ).
% list.pred_True
thf(fact_6984_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F2: A > B,G2: A > B] :
( ( X = Ya )
=> ( ( list_all @ A
@ ^ [Z5: A] :
( ( F2 @ Z5 )
= ( G2 @ Z5 ) )
@ Ya )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G2 @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_6985_implode_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ char ) @ ( list @ char )
@ ^ [Y5: list @ char,Z4: list @ char] : Y5 = Z4
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( map @ char @ char @ ascii_of )
@ ( map @ char @ char @ ascii_of ) ) ).
% implode.rsp
thf(fact_6986_list__ex1__simps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_ex1 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( ( P @ X )
=> ( list_all @ A
@ ^ [Y3: A] :
( ~ ( P @ Y3 )
| ( X = Y3 ) )
@ Xs ) )
& ( ~ ( P @ X )
=> ( list_ex1 @ A @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_6987_literal__of__asciis_Orsp,axiom,
( bNF_rel_fun @ ( list @ code_integer ) @ ( list @ code_integer ) @ ( list @ char ) @ ( list @ char )
@ ^ [Y5: list @ code_integer,Z4: list @ code_integer] : Y5 = Z4
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) )
@ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) ) ) ).
% literal_of_asciis.rsp
thf(fact_6988_Literal_Oabs__eq,axiom,
! [X: list @ char,Xg: $o,Xf: $o,Xe: $o,Xd: $o,Xc: $o,Xb: $o,Xa: $o] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X3: char] :
( ( member @ char @ X3 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( literal2 @ Xg @ Xf @ Xe @ Xd @ Xc @ Xb @ Xa @ ( abs_literal @ X ) )
= ( abs_literal @ ( cons @ char @ ( char2 @ Xg @ Xf @ Xe @ Xd @ Xc @ Xb @ Xa @ $false ) @ X ) ) ) ) ).
% Literal.abs_eq
thf(fact_6989_String_OLiteral__def,axiom,
( literal2
= ( map_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > ( list @ char ) > ( list @ char ) ) @ ( $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( ( list @ char ) > ( list @ char ) ) @ ( literal > literal ) @ ( id @ $o ) @ ( map_fun @ literal @ ( list @ char ) @ ( list @ char ) @ literal @ explode @ abs_literal ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) ) ) ) ).
% String.Literal_def
thf(fact_6990_Literal_Otransfer,axiom,
( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > $o > literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( $o > ( list @ char ) > ( list @ char ) ) @ ( $o > literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ $o @ $o @ ( ( list @ char ) > ( list @ char ) ) @ ( literal > literal )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( bNF_rel_fun @ ( list @ char ) @ literal @ ( list @ char ) @ literal @ pcr_literal @ pcr_literal ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) )
@ literal2 ) ).
% Literal.transfer
thf(fact_6991_listsp__inf__eq,axiom,
! [A: $tType,A4: A > $o,B3: A > $o] :
( ( listsp @ A @ ( inf_inf @ ( A > $o ) @ A4 @ B3 ) )
= ( inf_inf @ ( ( list @ A ) > $o ) @ ( listsp @ A @ A4 ) @ ( listsp @ A @ B3 ) ) ) ).
% listsp_inf_eq
thf(fact_6992_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% inv_o_cancel
thf(fact_6993_listsp__conj__eq,axiom,
! [A: $tType,A4: A > $o,B3: A > $o] :
( ( listsp @ A
@ ^ [X3: A] :
( ( A4 @ X3 )
& ( B3 @ X3 ) ) )
= ( ^ [X3: list @ A] :
( ( listsp @ A @ A4 @ X3 )
& ( listsp @ A @ B3 @ X3 ) ) ) ) ).
% listsp_conj_eq
thf(fact_6994_inv__identity,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [A5: A] : A5 )
= ( ^ [A5: A] : A5 ) ) ).
% inv_identity
thf(fact_6995_inv__id,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
= ( id @ A ) ) ).
% inv_id
thf(fact_6996_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B,G2: A > C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ F2 )
= G2 ) ) ).
% o_inv_o_cancel
thf(fact_6997_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F2: A > B,L: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( map @ A @ B @ F2 @ L ) )
= L ) ) ).
% inj_map_inv_f
thf(fact_6998_listsp__infI,axiom,
! [A: $tType,A4: A > $o,L: list @ A,B3: A > $o] :
( ( listsp @ A @ A4 @ L )
=> ( ( listsp @ A @ B3 @ L )
=> ( listsp @ A @ ( inf_inf @ ( A > $o ) @ A4 @ B3 ) @ L ) ) ) ).
% listsp_infI
thf(fact_6999_bij__imp__bij__inv,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_imp_bij_inv
thf(fact_7000_bij__inv__eq__iff,axiom,
! [A: $tType,B: $tType,P4: A > B,X: A,Y: B] :
( ( bij_betw @ A @ B @ P4 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( X
= ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P4 @ Y ) )
= ( ( P4 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_7001_inv__inv__eq,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
= F2 ) ) ).
% inv_inv_eq
thf(fact_7002_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F2: B > A,Y: A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_7003_surj__iff__all,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X3: A] :
( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_7004_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F2 @ ( image2 @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ A4 ) )
= A4 ) ) ).
% image_f_inv_f
thf(fact_7005_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F2: B > A,G2: A > B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X2: B] :
( ( G2 @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= G2 ) ) ) ).
% surj_imp_inv_eq
thf(fact_7006_inv__def,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= ( ^ [Y3: A] :
( fChoice @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= Y3 ) ) ) ) ).
% inv_def
thf(fact_7007_inv__equality,axiom,
! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
( ! [X2: A] :
( ( G2 @ ( F2 @ X2 ) )
= X2 )
=> ( ! [Y2: B] :
( ( F2 @ ( G2 @ Y2 ) )
= Y2 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= G2 ) ) ) ).
% inv_equality
thf(fact_7008_inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X ) )
= X ) ) ).
% inv_f_f
thf(fact_7009_inv__f__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Y: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X )
= Y )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y )
= X ) ) ) ).
% inv_f_eq
thf(fact_7010_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,G2: B > A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ! [X2: B] :
( ( F2 @ ( G2 @ X2 ) )
= X2 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= G2 ) ) ) ).
% inj_imp_inv_eq
thf(fact_7011_lists__def,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A6: set @ A] :
( collect @ ( list @ A )
@ ( listsp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A6 ) ) ) ) ) ).
% lists_def
thf(fact_7012_listsp__lists__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( listsp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( ^ [X3: list @ A] : ( member @ ( list @ A ) @ X3 @ ( lists @ A @ A4 ) ) ) ) ).
% listsp_lists_eq
thf(fact_7013_inv__into__def2,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A6: set @ A,F: A > B,X3: B] :
( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( ( F @ Y3 )
= X3 ) ) ) ) ) ).
% inv_into_def2
thf(fact_7014_inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A6: set @ A,F: A > B,X3: B] :
( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( ( F @ Y3 )
= X3 ) ) ) ) ) ).
% inv_into_def
thf(fact_7015_inj__transfer,axiom,
! [B: $tType,A: $tType,F2: A > B,P: A > $o,X: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ! [Y2: B] :
( ( member @ B @ Y2 @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_7016_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( image2 @ A @ B @ F2 @ A4 ) )
= A4 ) ) ).
% image_inv_f_f
thf(fact_7017_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% inj_imp_surj_inv
thf(fact_7018_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_inj_inv
thf(fact_7019_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F2: B > A,G2: A > B] :
( ( ( comp @ B @ A @ A @ F2 @ G2 )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G2 @ F2 )
= ( id @ B ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= G2 ) ) ) ).
% inv_unique_comp
thf(fact_7020_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G2: C > A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F2 @ G2 ) )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G2 ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% o_inv_distrib
thf(fact_7021_inv__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( compow @ ( A > A ) @ N @ F2 ) )
= ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ).
% inv_fn
thf(fact_7022_mono__inv,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B] :
( ( order_mono @ A @ B @ F2 )
=> ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( order_mono @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% mono_inv
thf(fact_7023_iso__backward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F2: B > A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
=> ( ( bNF_Wellorder_iso @ B @ A @ R3 @ R5 @ F2 )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R3 ) @ F2 @ X ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R3 ) @ F2 @ Y ) ) @ R3 ) ) ) ).
% iso_backward
thf(fact_7024_bij__image__Collect__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,P: A > $o] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( collect @ A @ P ) )
= ( collect @ B
@ ^ [Y3: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y3 ) ) ) ) ) ).
% bij_image_Collect_eq
thf(fact_7025_from__nat__def,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( from_nat @ A )
= ( hilbert_inv_into @ A @ nat @ ( top_top @ ( set @ A ) ) @ ( to_nat @ A ) ) ) ) ).
% from_nat_def
thf(fact_7026_surj__iff,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) )
= ( id @ A ) ) ) ).
% surj_iff
thf(fact_7027_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F2: A > B,M4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( image2 @ A @ B @ F2 @ M4 ) @ M4 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_7028_inj__iff,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% inj_iff
thf(fact_7029_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( vimage @ A @ B @ F2 @ A4 )
= ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ A4 ) ) ) ).
% bij_vimage_eq_inv_image
thf(fact_7030_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) )
= ( ^ [X3: A] : X3 ) ) ) ).
% fn_o_inv_fn_is_id
thf(fact_7031_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ ( compow @ ( A > A ) @ N @ F2 ) )
= ( ^ [X3: A] : X3 ) ) ) ).
% inv_fn_o_fn_is_id
thf(fact_7032_bijection_Oinv__comp__left,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% bijection.inv_comp_left
thf(fact_7033_bijection_Oinv__comp__right,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( comp @ A @ A @ A @ F2 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) )
= ( id @ A ) ) ) ).
% bijection.inv_comp_right
thf(fact_7034_bijection_Osurj,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj
thf(fact_7035_bijection_Oinj,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj
thf(fact_7036_bijection__def,axiom,
! [A: $tType] :
( ( hilbert_bijection @ A )
= ( ^ [F: A > A] : ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bijection_def
thf(fact_7037_bijection_Ointro,axiom,
! [A: $tType,F2: A > A] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( hilbert_bijection @ A @ F2 ) ) ).
% bijection.intro
thf(fact_7038_bijection_Obij,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij
thf(fact_7039_bijection_Oeq__invI,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eq_invI
thf(fact_7040_bijection_Oinv__left,axiom,
! [A: $tType,F2: A > A,A3: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ A3 ) )
= A3 ) ) ).
% bijection.inv_left
thf(fact_7041_bijection_Oinv__right,axiom,
! [A: $tType,F2: A > A,A3: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( F2 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 ) )
= A3 ) ) ).
% bijection.inv_right
thf(fact_7042_bijection_Oeq__inv__iff,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_inv_iff
thf(fact_7043_bijection_Oinv__left__eq__iff,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 )
= B2 )
= ( ( F2 @ B2 )
= A3 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_7044_bijection_Oinv__right__eq__iff,axiom,
! [A: $tType,F2: A > A,B2: A,A3: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( B2
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 ) )
= ( ( F2 @ B2 )
= A3 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_7045_bijection_Osurj__inv,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( image2 @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj_inv
thf(fact_7046_bijection_Oinj__inv,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj_inv
thf(fact_7047_bijection_Obij__inv,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij_inv
thf(fact_7048_cr__integer__def,axiom,
( code_cr_integer
= ( ^ [X3: int,Y3: code_integer] :
( X3
= ( code_int_of_integer @ Y3 ) ) ) ) ).
% cr_integer_def
thf(fact_7049_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_7050_ndepth__Push__Node__aux,axiom,
! [A: $tType,I: nat,F2: nat > ( sum_sum @ A @ nat ),K: nat] :
( ( ( case_nat @ ( sum_sum @ A @ nat ) @ ( sum_Inr @ nat @ A @ ( suc @ I ) ) @ F2 @ K )
= ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) )
=> ( ord_less_eq @ nat
@ ( suc
@ ( ord_Least @ nat
@ ^ [X3: nat] :
( ( F2 @ X3 )
= ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) )
@ K ) ) ).
% ndepth_Push_Node_aux
thf(fact_7051_Powp__Pow__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( powp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( ^ [X3: set @ A] : ( member @ ( set @ A ) @ X3 @ ( pow2 @ A @ A4 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_7052_Powp__def,axiom,
! [A: $tType] :
( ( powp @ A )
= ( ^ [A6: A > $o,B5: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( A6 @ X3 ) ) ) ) ).
% Powp_def
thf(fact_7053_Union__sum,axiom,
! [C: $tType,A: $tType,B: $tType,F2: ( sum_sum @ A @ B ) > ( set @ C )] :
( ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ ( sum_sum @ A @ B ) @ ( set @ C ) @ F2 @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) )
= ( sup_sup @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [L2: A] : ( F2 @ ( sum_Inl @ A @ B @ L2 ) )
@ ( top_top @ ( set @ A ) ) ) )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ B @ ( set @ C )
@ ^ [R4: B] : ( F2 @ ( sum_Inr @ B @ A @ R4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% Union_sum
thf(fact_7054_Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Node @ B @ A )
= ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
@ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
? [F: nat > ( sum_sum @ B @ nat ),X3: sum_sum @ A @ nat,K4: nat] :
( ( P6
= ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F @ X3 ) )
& ( ( F @ K4 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% Node_def
thf(fact_7055_Plus__def,axiom,
! [B: $tType,A: $tType] :
( ( sum_Plus @ A @ B )
= ( ^ [A6: set @ A,B5: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image2 @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 ) @ ( image2 @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B5 ) ) ) ) ).
% Plus_def
thf(fact_7056_UNIV__sum,axiom,
! [A: $tType,B: $tType] :
( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
= ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image2 @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image2 @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% UNIV_sum
thf(fact_7057_Node__K0__I,axiom,
! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
@ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
@ ^ [K4: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
@ A3 )
@ ( old_Node @ A @ B ) ) ).
% Node_K0_I
thf(fact_7058_Field__csum,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R3 @ S3 ) )
= ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image2 @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( field2 @ A @ R3 ) ) @ ( image2 @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( field2 @ B @ S3 ) ) ) ) ).
% Field_csum
thf(fact_7059_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B3: set @ C,G2: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G2 @ ( sum_Plus @ B @ C @ A4 @ B3 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G2 @ ( sum_Inl @ B @ C ) ) @ A4 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G2 @ ( sum_Inr @ C @ B ) ) @ B3 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_7060_Union__plus,axiom,
! [A: $tType,B: $tType,C: $tType,F2: ( sum_sum @ B @ C ) > ( set @ A ),A4: set @ B,B3: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( sum_sum @ B @ C ) @ ( set @ A ) @ F2 @ ( sum_Plus @ B @ C @ A4 @ B3 ) ) )
= ( sup_sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [A5: B] : ( F2 @ ( sum_Inl @ B @ C @ A5 ) )
@ A4 ) )
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [B4: C] : ( F2 @ ( sum_Inr @ C @ B @ B4 ) )
@ B3 ) ) ) ) ).
% Union_plus
thf(fact_7061_int__encode__def,axiom,
( nat_int_encode
= ( ^ [I3: int] : ( nat_sum_encode @ ( if @ ( sum_sum @ nat @ nat ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 ) @ ( sum_Inl @ nat @ nat @ ( nat2 @ I3 ) ) @ ( sum_Inr @ nat @ nat @ ( nat2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ I3 ) @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% int_encode_def
thf(fact_7062_sum__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A] :
( ( basic_setl @ A @ B @ ( sum_Inl @ A @ B @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(1)
thf(fact_7063_sum__set__simps_I2_J,axiom,
! [A: $tType,C: $tType,X: A] :
( ( basic_setl @ C @ A @ ( sum_Inr @ A @ C @ X ) )
= ( bot_bot @ ( set @ C ) ) ) ).
% sum_set_simps(2)
thf(fact_7064_surj__sum__encode,axiom,
( ( image2 @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_sum_encode
thf(fact_7065_bij__sum__encode,axiom,
bij_betw @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_sum_encode
thf(fact_7066_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ^ [B4: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B4 ) ) ) ) ).
% sum_encode_def
thf(fact_7067_case__sum__KK,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C] :
( ( sum_case_sum @ A @ C @ B
@ ^ [X3: A] : A3
@ ^ [X3: B] : A3 )
= ( ^ [X3: sum_sum @ A @ B] : A3 ) ) ).
% case_sum_KK
thf(fact_7068_surj__sum__decode,axiom,
( ( image2 @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ) ).
% surj_sum_decode
thf(fact_7069_bij__sum__decode,axiom,
bij_betw @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ).
% bij_sum_decode
thf(fact_7070_sum_Ocase__distrib,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,H3: C > D,F1: A > C,F22: B > C,Sum: sum_sum @ A @ B] :
( ( H3 @ ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ Sum ) )
= ( sum_case_sum @ A @ D @ B
@ ^ [X3: A] : ( H3 @ ( F1 @ X3 ) )
@ ^ [X3: B] : ( H3 @ ( F22 @ X3 ) )
@ Sum ) ) ).
% sum.case_distrib
thf(fact_7071_disjE__realizer,axiom,
! [C: $tType,B: $tType,A: $tType,P: A > $o,Q2: B > $o,X: sum_sum @ A @ B,R: C > $o,F2: A > C,G2: B > C] :
( ( sum_case_sum @ A @ $o @ B @ P @ Q2 @ X )
=> ( ! [P11: A] :
( ( P @ P11 )
=> ( R @ ( F2 @ P11 ) ) )
=> ( ! [Q7: B] :
( ( Q2 @ Q7 )
=> ( R @ ( G2 @ Q7 ) ) )
=> ( R @ ( sum_case_sum @ A @ C @ B @ F2 @ G2 @ X ) ) ) ) ) ).
% disjE_realizer
thf(fact_7072_surjective__sum,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( sum_sum @ A @ B ) > C] :
( ( sum_case_sum @ A @ C @ B
@ ^ [X3: A] : ( F2 @ ( sum_Inl @ A @ B @ X3 ) )
@ ^ [Y3: B] : ( F2 @ ( sum_Inr @ B @ A @ Y3 ) ) )
= F2 ) ).
% surjective_sum
thf(fact_7073_nth__item_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nth_item_rel @ A0 )
=> ( ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) )
=> ( ! [A15: nat,Aa4: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl @ nat @ nat @ A15 ) )
=> ( ( ( nat_sum_decode @ A15 )
= ( sum_Inl @ nat @ nat @ Aa4 ) )
=> ( P @ Aa4 ) ) )
=> ( ! [A15: nat,B13: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl @ nat @ nat @ A15 ) )
=> ( ( ( nat_sum_decode @ A15 )
= ( sum_Inr @ nat @ nat @ B13 ) )
=> ( P @ B13 ) ) )
=> ( ! [B13: nat,Ba2: nat,X5: nat,Y6: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr @ nat @ nat @ B13 ) )
=> ( ( ( nat_sum_decode @ B13 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X5 @ Y6 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P @ X5 ) ) ) )
=> ( ! [B13: nat,Ba2: nat,X5: nat,Y6: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr @ nat @ nat @ B13 ) )
=> ( ( ( nat_sum_decode @ B13 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X5 @ Y6 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P @ Y6 ) ) ) )
=> ( P @ ( suc @ N3 ) ) ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% nth_item.pinduct
thf(fact_7074_int__decode__def,axiom,
( nat_int_decode
= ( ^ [N2: nat] :
( sum_case_sum @ nat @ int @ nat @ ( semiring_1_of_nat @ int )
@ ^ [B4: nat] : ( minus_minus @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ B4 ) ) @ ( one_one @ int ) )
@ ( nat_sum_decode @ N2 ) ) ) ) ).
% int_decode_def
thf(fact_7075_sum__set__defs_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( basic_setl @ A @ B )
= ( sum_case_sum @ A @ ( set @ A ) @ B
@ ^ [Z5: A] : ( insert2 @ A @ Z5 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [B4: B] : ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_defs(1)
thf(fact_7076_sum__set__simps_I4_J,axiom,
! [E: $tType,A: $tType,X: A] :
( ( basic_setr @ E @ A @ ( sum_Inr @ A @ E @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(4)
thf(fact_7077_card__order__csum__cone__cexp__def,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A18: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ( bNF_Cardinal_cexp @ ( sum_sum @ B @ product_unit ) @ A @ ( bNF_Cardinal_csum @ B @ product_unit @ ( bNF_Ca6860139660246222851ard_of @ B @ A18 ) @ bNF_Cardinal_cone ) @ R3 )
= ( bNF_Ca6860139660246222851ard_of @ ( A > ( sum_sum @ B @ product_unit ) ) @ ( bNF_Wellorder_Func @ A @ ( sum_sum @ B @ product_unit ) @ ( top_top @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( sum_sum @ B @ product_unit ) ) @ ( image2 @ B @ ( sum_sum @ B @ product_unit ) @ ( sum_Inl @ B @ product_unit ) @ A18 ) @ ( insert2 @ ( 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
thf(fact_7078_unit__abs__eta__conv,axiom,
! [A: $tType,F2: product_unit > A] :
( ( ^ [U2: product_unit] : ( F2 @ product_Unity ) )
= F2 ) ).
% unit_abs_eta_conv
thf(fact_7079_sum__set__simps_I3_J,axiom,
! [A: $tType,D: $tType,X: A] :
( ( basic_setr @ A @ D @ ( sum_Inl @ A @ D @ X ) )
= ( bot_bot @ ( set @ D ) ) ) ).
% sum_set_simps(3)
thf(fact_7080_sup__unit__def,axiom,
( ( sup_sup @ product_unit )
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_7081_cone__def,axiom,
( bNF_Cardinal_cone
= ( bNF_Ca6860139660246222851ard_of @ product_unit @ ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ) ).
% cone_def
thf(fact_7082_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_7083_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_7084_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_7085_top__unit__def,axiom,
( ( top_top @ product_unit )
= product_Unity ) ).
% top_unit_def
thf(fact_7086_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_7087_sum__set__defs_I2_J,axiom,
! [C: $tType,D: $tType] :
( ( basic_setr @ C @ D )
= ( sum_case_sum @ C @ ( set @ D ) @ D
@ ^ [A5: C] : ( bot_bot @ ( set @ D ) )
@ ^ [Z5: D] : ( insert2 @ D @ Z5 @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% sum_set_defs(2)
thf(fact_7088_Push__def,axiom,
! [B: $tType] :
( ( old_Push @ B )
= ( case_nat @ ( sum_sum @ B @ nat ) ) ) ).
% Push_def
thf(fact_7089_natural__zero__minus__one,axiom,
( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
= ( zero_zero @ code_natural ) ) ).
% natural_zero_minus_one
thf(fact_7090_log_Oelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_7091_log_Osimps,axiom,
( log
= ( ^ [B4: code_natural,I3: code_natural] :
( if @ code_natural
@ ( ( ord_less_eq @ code_natural @ B4 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ I3 @ B4 ) )
@ ( one_one @ code_natural )
@ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B4 @ ( divide_divide @ code_natural @ I3 @ B4 ) ) ) ) ) ) ).
% log.simps
thf(fact_7092_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K4: code_natural,F: B > A > ( product_prod @ B @ A ),X3: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K4
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X3 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F @ X3 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K4 @ ( one_one @ code_natural ) ) @ F ) ) ) ) ) ).
% iterate.simps
thf(fact_7093_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_7094_Random_Orange__def,axiom,
( range
= ( ^ [K4: code_natural] :
( 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 ) )
@ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
@ ^ [L2: code_natural] :
( 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
@ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V2 @ ( times_times @ code_natural @ L2 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V2 @ K4 ) ) ) ) ) ).
% Random.range_def
thf(fact_7095_next_Osimps,axiom,
! [V: code_natural,W2: code_natural] :
( ( next @ ( product_Pair @ code_natural @ code_natural @ V @ W2 ) )
= ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_7096_log_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
~ ! [B7: code_natural,I2: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ B7 @ I2 ) ) ).
% log.cases
thf(fact_7097_full__exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F3: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_7098_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc3360725361186068524ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc3360725361186068524ustive @ A ) )
=> ! [X: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F3: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% full_exhaustive_fun'.cases
thf(fact_7099_exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F3: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
( X
!= ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_7100_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc658316121487927005ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc658316121487927005ustive @ A ) )
=> ! [X: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F3: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% exhaustive_fun'.cases
thf(fact_7101_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F3: code_natural > A,N3: code_natural,M3: code_natural] :
( X
!= ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M3 ) ) ) ).
% Lazy_Sequence.iterate_upto.cases
thf(fact_7102_iter_Ocases,axiom,
! [A: $tType,X: product_prod @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )] :
~ ! [Random: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ),Nrandom: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Random @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Nrandom @ Seed ) ) ) ).
% iter.cases
thf(fact_7103_split__seed__def,axiom,
( split_seed
= ( ^ [S2: 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 ) )
@ ^ [V2: code_natural,W3: code_natural] :
( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [V6: code_natural,W4: code_natural] : ( product_Pair @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_Pair @ code_natural @ code_natural @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W4 ) @ ( product_Pair @ code_natural @ code_natural @ V6 @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W3 ) ) )
@ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S2 ) ) )
@ S2 ) ) ) ).
% split_seed_def
thf(fact_7104_inc__shift__def,axiom,
( inc_shift
= ( ^ [V2: code_natural,K4: code_natural] : ( if @ code_natural @ ( V2 = K4 ) @ ( one_one @ code_natural ) @ ( plus_plus @ code_natural @ K4 @ ( one_one @ code_natural ) ) ) ) ) ).
% inc_shift_def
thf(fact_7105_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: code_natural > A,A1: code_natural,A22: code_natural,P: ( code_natural > A ) > code_natural > code_natural > $o] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ A0 @ ( product_Pair @ code_natural @ code_natural @ A1 @ A22 ) ) )
=> ( ! [F3: code_natural > A,N3: code_natural,M3: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M3 ) ) )
=> ( ! [X5: product_unit] :
( ~ ( ord_less @ code_natural @ M3 @ N3 )
=> ( P @ F3 @ ( plus_plus @ code_natural @ N3 @ ( one_one @ code_natural ) ) @ M3 ) )
=> ( P @ F3 @ N3 @ M3 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_7106_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_7107_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
=> ~ ( ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_7108_pick__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( pick @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K4: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K4 ) )
@ Xs ) )
= ( pick @ A @ Xs ) ) ).
% pick_drop_zero
thf(fact_7109_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
~ ! [K2: code_natural,F3: B > A > ( product_prod @ B @ A ),X2: B] :
( X
!= ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F3 @ X2 ) ) ) ).
% iterate.cases
thf(fact_7110_select__weight__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K4: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K4 ) )
@ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_drop_zero
thf(fact_7111_select__weight__def,axiom,
! [A: $tType] :
( ( select_weight @ A )
= ( ^ [Xs2: list @ ( product_prod @ code_natural @ A )] :
( 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 @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs2 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs2 @ K4 ) ) ) ) ) ).
% select_weight_def
thf(fact_7112_select__weight__cons__zero,axiom,
! [A: $tType,X: A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X ) @ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_cons_zero
thf(fact_7113_select__weight__select,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) )
= ( select @ A @ Xs ) ) ) ).
% select_weight_select
thf(fact_7114_pick__same,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs @ L ) ) ) ).
% pick_same
thf(fact_7115_one__natural__def,axiom,
( ( one_one @ code_natural )
= ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% one_natural_def
thf(fact_7116_natural__of__nat__cases,axiom,
! [X: code_natural] :
~ ! [Y2: nat] :
( ( X
= ( code_natural_of_nat @ Y2 ) )
=> ~ ( member @ nat @ Y2 @ ( top_top @ ( set @ nat ) ) ) ) ).
% natural_of_nat_cases
thf(fact_7117_natural__of__nat__induct,axiom,
! [P: code_natural > $o,X: code_natural] :
( ! [Y2: nat] :
( ( member @ nat @ Y2 @ ( top_top @ ( set @ nat ) ) )
=> ( P @ ( code_natural_of_nat @ Y2 ) ) )
=> ( P @ X ) ) ).
% natural_of_nat_induct
thf(fact_7118_natural__of__nat__inject,axiom,
! [X: nat,Y: nat] :
( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
=> ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ( ( code_natural_of_nat @ X )
= ( code_natural_of_nat @ Y ) )
= ( X = Y ) ) ) ) ).
% natural_of_nat_inject
thf(fact_7119_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs2: list @ A] :
( 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 @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs2 @ ( code_nat_of_natural @ K4 ) ) ) ) ) ) ).
% select_def
thf(fact_7120_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 ) ) )] :
~ ! [T6: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T6 @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Nrandom @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Sz @ Seed ) ) ) ) ) ).
% iter'.cases
thf(fact_7121_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( one_one @ code_natural ) )
= ( one_one @ nat ) ) ).
% one_natural.rep_eq
thf(fact_7122_type__definition__natural,axiom,
type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% type_definition_natural
thf(fact_7123_natural__of__nat__inverse,axiom,
! [Y: nat] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
= Y ) ) ).
% natural_of_nat_inverse
thf(fact_7124_nat__of__natural__induct,axiom,
! [Y: nat,P: nat > $o] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X2: code_natural] : ( P @ ( code_nat_of_natural @ X2 ) )
=> ( P @ Y ) ) ) ).
% nat_of_natural_induct
thf(fact_7125_nat__of__natural__cases,axiom,
! [Y: nat] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ~ ! [X2: code_natural] :
( Y
!= ( code_nat_of_natural @ X2 ) ) ) ).
% nat_of_natural_cases
thf(fact_7126_nat__of__natural,axiom,
! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% nat_of_natural
thf(fact_7127_cr__natural__def,axiom,
( code_cr_natural
= ( ^ [X3: nat,Y3: code_natural] :
( X3
= ( code_nat_of_natural @ Y3 ) ) ) ) ).
% cr_natural_def
thf(fact_7128_ndepth__def,axiom,
! [B: $tType,A: $tType] :
( ( old_ndepth @ A @ B )
= ( ^ [N2: old_node @ A @ B] :
( product_case_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ nat
@ ^ [F: nat > ( sum_sum @ B @ nat ),X3: sum_sum @ A @ nat] :
( ord_Least @ nat
@ ^ [K4: nat] :
( ( F @ K4 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) )
@ ( old_Rep_Node @ A @ B @ N2 ) ) ) ) ).
% ndepth_def
thf(fact_7129_ndepth__K0,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
( ( old_ndepth @ A @ B
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X ) ) )
= ( zero_zero @ nat ) ) ).
% ndepth_K0
thf(fact_7130_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) ) ) ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X13: A,X24: A,Xs3: list @ A] :
( ( Xa
= ( cons @ A @ X13 @ ( cons @ A @ X24 @ Xs3 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X24 @ Xs24 ) ) @ Xs3 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X24 @ Xs3 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_7131_Atom__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Atom @ A @ B )
= ( ^ [X3: sum_sum @ A @ nat] :
( insert2 @ ( old_node @ A @ B )
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X3 ) )
@ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% Atom_def
thf(fact_7132_Push__Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Push_Node @ B @ A )
= ( ^ [N2: sum_sum @ B @ nat,X3: old_node @ A @ B] : ( old_Abs_Node @ B @ A @ ( product_apfst @ ( nat > ( sum_sum @ B @ nat ) ) @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ ( old_Push @ B @ N2 ) @ ( old_Rep_Node @ A @ B @ X3 ) ) ) ) ) ).
% Push_Node_def
thf(fact_7133_inj__Atom,axiom,
! [B: $tType,A: $tType] : ( inj_on @ ( sum_sum @ A @ nat ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Atom @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ nat ) ) ) ) ).
% inj_Atom
thf(fact_7134_Lim__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Lim @ B @ A )
= ( ^ [F: B > ( set @ ( old_node @ A @ B ) )] :
( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( collect @ ( set @ ( old_node @ A @ B ) )
@ ^ [Z5: set @ ( old_node @ A @ B )] :
? [X3: B] :
( Z5
= ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inl @ B @ nat @ X3 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% Lim_def
thf(fact_7135_Scons__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Scons @ A @ B )
= ( ^ [M9: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M9 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N11 ) ) ) ) ).
% Scons_def
thf(fact_7136_Scons__UN1__y,axiom,
! [A: $tType,B: $tType,C: $tType,M4: set @ ( old_node @ A @ B ),F2: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_Scons @ A @ B @ M4 @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F2 @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X3: C] : ( old_Scons @ A @ B @ M4 @ ( F2 @ X3 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% Scons_UN1_y
thf(fact_7137_Scons__UN1__x,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > ( set @ ( old_node @ A @ B ) ),M4: set @ ( old_node @ A @ B )] :
( ( old_Scons @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F2 @ ( top_top @ ( set @ C ) ) ) ) @ M4 )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X3: C] : ( old_Scons @ A @ B @ ( F2 @ X3 ) @ M4 )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% Scons_UN1_x
thf(fact_7138_Split__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( old_Split @ A @ B @ C )
= ( ^ [C5: ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > C,M9: set @ ( old_node @ A @ B )] :
( the @ C
@ ^ [U2: C] :
? [X3: set @ ( old_node @ A @ B ),Y3: set @ ( old_node @ A @ B )] :
( ( M9
= ( old_Scons @ A @ B @ X3 @ Y3 ) )
& ( U2
= ( C5 @ X3 @ Y3 ) ) ) ) ) ) ).
% Split_def
thf(fact_7139_inj__Leaf,axiom,
! [B: $tType,A: $tType] : ( inj_on @ A @ ( set @ ( old_node @ A @ B ) ) @ ( old_Leaf @ A @ B ) @ ( top_top @ ( set @ A ) ) ) ).
% inj_Leaf
thf(fact_7140_ntrunc__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,K: nat,F2: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_ntrunc @ A @ B @ K @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F2 @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X3: C] : ( old_ntrunc @ A @ B @ K @ ( F2 @ X3 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% ntrunc_UN1
thf(fact_7141_In1__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_In1 @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F2 @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X3: C] : ( old_In1 @ A @ B @ ( F2 @ X3 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% In1_UN1
thf(fact_7142_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
thf(fact_7143_ntrunc__one__In1,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In1 @ A @ B @ M4 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In1
thf(fact_7144_inj__In1,axiom,
! [B: $tType,A: $tType] : ( inj_on @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B ) @ ( top_top @ ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
% inj_In1
thf(fact_7145_ntrunc__def,axiom,
! [A: $tType,B: $tType] :
( ( old_ntrunc @ A @ B )
= ( ^ [K4: nat,N11: set @ ( old_node @ A @ B )] :
( collect @ ( old_node @ A @ B )
@ ^ [N2: old_node @ A @ B] :
( ( member @ ( old_node @ A @ B ) @ N2 @ N11 )
& ( ord_less @ nat @ ( old_ndepth @ A @ B @ N2 ) @ K4 ) ) ) ) ) ).
% ntrunc_def
thf(fact_7146_ntrunc__one__In0,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In0 @ A @ B @ M4 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In0
thf(fact_7147_In1__def,axiom,
! [A: $tType,B: $tType] :
( ( old_In1 @ A @ B )
= ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% In1_def
thf(fact_7148_inj__Numb,axiom,
! [A: $tType,B: $tType] : ( inj_on @ nat @ ( set @ ( old_node @ A @ B ) ) @ ( old_Numb @ A @ B ) @ ( top_top @ ( set @ nat ) ) ) ).
% inj_Numb
thf(fact_7149_inj__In0,axiom,
! [B: $tType,A: $tType] : ( inj_on @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B ) @ ( top_top @ ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
% inj_In0
thf(fact_7150_In0__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_In0 @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F2 @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X3: C] : ( old_In0 @ A @ B @ ( F2 @ X3 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% In0_UN1
thf(fact_7151_Case__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( old_Case @ A @ B @ C )
= ( ^ [C5: ( set @ ( old_node @ A @ B ) ) > C,D5: ( set @ ( old_node @ A @ B ) ) > C,M9: set @ ( old_node @ A @ B )] :
( the @ C
@ ^ [U2: C] :
( ? [X3: set @ ( old_node @ A @ B )] :
( ( M9
= ( old_In0 @ A @ B @ X3 ) )
& ( U2
= ( C5 @ X3 ) ) )
| ? [Y3: set @ ( old_node @ A @ B )] :
( ( M9
= ( old_In1 @ A @ B @ Y3 ) )
& ( U2
= ( D5 @ Y3 ) ) ) ) ) ) ) ).
% Case_def
thf(fact_7152_nth__item_Opelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X: nat,Y: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X )
= Y )
=> ( ( accp @ nat @ nth_item_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A5: nat,B4: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B4 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% nth_item.pelims
thf(fact_7153_nth__item_Osimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A5: nat,B4: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B4 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N ) ) ) ) ).
% nth_item.simps(2)
thf(fact_7154_nth__item_Oelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X: nat,Y: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A5: nat,B4: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B4 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N3 ) ) ) ) ) ) ) ).
% nth_item.elims
thf(fact_7155_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
=> ( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A5: nat,B4: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B4 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N ) ) ) ) ) ).
% nth_item.psimps(2)
thf(fact_7156_dsum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dsum @ A @ B )
= ( ^ [R4: 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 ) ) )] :
( sup_sup @ ( 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 ) ) ) )
@ ( 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 ) ) ) )
@ ( 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 ) ) ) )
@ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] : ( insert2 @ ( 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 @ X9 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ R4 ) )
@ ( complete_Sup_Sup @ ( 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 ) ) ) )
@ ( 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 ) ) ) )
@ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( 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 @ Y3 ) @ ( old_In1 @ A @ B @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S2 ) ) ) ) ) ).
% dsum_def
thf(fact_7157_dsumE,axiom,
! [B: $tType,A: $tType,W2: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S3: 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 @ R3 @ S3 ) )
=> ( ! [X2: set @ ( old_node @ A @ B ),X11: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X11 ) @ R3 )
=> ( W2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X2 ) @ ( old_In0 @ A @ B @ X11 ) ) ) )
=> ~ ! [Y2: set @ ( old_node @ A @ B ),Y10: set @ ( old_node @ A @ B )] :
( ( member @ ( 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 @ Y10 ) @ S3 )
=> ( W2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y2 ) @ ( old_In1 @ A @ B @ Y10 ) ) ) ) ) ) ).
% dsumE
thf(fact_7158_dsum__In0I,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B ),M11: set @ ( old_node @ A @ B ),R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S3: 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 ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M4 @ M11 ) @ R3 )
=> ( member @ ( 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 @ M11 ) ) @ ( old_dsum @ A @ B @ R3 @ S3 ) ) ) ).
% dsum_In0I
thf(fact_7159_dsum__In1I,axiom,
! [B: $tType,A: $tType,N4: set @ ( old_node @ A @ B ),N12: set @ ( old_node @ A @ B ),S3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),R3: 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 ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N4 @ N12 ) @ S3 )
=> ( member @ ( 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 @ N4 ) @ ( old_In1 @ A @ B @ N12 ) ) @ ( old_dsum @ A @ B @ R3 @ S3 ) ) ) ).
% dsum_In1I
thf(fact_7160_dprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dprod @ A @ B )
= ( ^ [R4: 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 ) ) )] :
( complete_Sup_Sup @ ( 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 ) ) ) )
@ ( 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 ) ) ) )
@ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] :
( complete_Sup_Sup @ ( 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 ) ) ) )
@ ( 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 ) ) ) )
@ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( 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 @ Y3 ) @ ( old_Scons @ A @ B @ X9 @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S2 ) ) )
@ R4 ) ) ) ) ).
% dprod_def
thf(fact_7161_dprodI,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B ),M11: set @ ( old_node @ A @ B ),R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),N4: set @ ( old_node @ A @ B ),N12: set @ ( old_node @ A @ B ),S3: 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 ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M4 @ M11 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N4 @ N12 ) @ S3 )
=> ( member @ ( 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 @ N4 ) @ ( old_Scons @ A @ B @ M11 @ N12 ) ) @ ( old_dprod @ A @ B @ R3 @ S3 ) ) ) ) ).
% dprodI
thf(fact_7162_dprodE,axiom,
! [B: $tType,A: $tType,C2: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S3: 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 @ R3 @ S3 ) )
=> ~ ! [X2: set @ ( old_node @ A @ B ),Y2: set @ ( old_node @ A @ B ),X11: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X11 ) @ R3 )
=> ! [Y10: set @ ( old_node @ A @ B )] :
( ( member @ ( 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 @ Y10 ) @ S3 )
=> ( C2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y2 ) @ ( old_Scons @ A @ B @ X11 @ Y10 ) ) ) ) ) ) ).
% dprodE
thf(fact_7163_usum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_usum @ A @ B )
= ( ^ [A6: set @ ( set @ ( old_node @ A @ B ) ),B5: set @ ( set @ ( old_node @ A @ B ) )] : ( sup_sup @ ( set @ ( set @ ( old_node @ A @ B ) ) ) @ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B ) @ A6 ) @ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B ) @ B5 ) ) ) ) ).
% usum_def
thf(fact_7164_MOST__INFM,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( frequently @ A @ P @ ( cofinite @ A ) ) ) ) ).
% MOST_INFM
thf(fact_7165_frequently__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( frequently @ A
@ ^ [X3: A] : P
@ F5 )
= P ) ) ).
% frequently_const
thf(fact_7166_not__MOST,axiom,
! [A: $tType,P: A > $o] :
( ( ~ ( eventually @ A @ P @ ( cofinite @ A ) ) )
= ( frequently @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ ( cofinite @ A ) ) ) ).
% not_MOST
thf(fact_7167_not__INFM,axiom,
! [A: $tType,P: A > $o] :
( ( ~ ( frequently @ A @ P @ ( cofinite @ A ) ) )
= ( eventually @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ ( cofinite @ A ) ) ) ).
% not_INFM
thf(fact_7168_INFM__neq_I2_J,axiom,
! [A: $tType,A3: A] :
( ( frequently @ A
@ ^ [X3: A] : A3 != X3
@ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% INFM_neq(2)
thf(fact_7169_INFM__neq_I1_J,axiom,
! [A: $tType,A3: A] :
( ( frequently @ A
@ ^ [X3: A] : X3 != A3
@ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% INFM_neq(1)
thf(fact_7170_INFM__const,axiom,
! [A: $tType,P: $o] :
( ( frequently @ A
@ ^ [X3: A] : P
@ ( cofinite @ A ) )
= ( P
& ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% INFM_const
thf(fact_7171_INFM__nat,axiom,
! [P: nat > $o] :
( ( frequently @ nat @ P @ ( cofinite @ nat ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat
thf(fact_7172_INFM__nat__le,axiom,
! [P: nat > $o] :
( ( frequently @ nat @ P @ ( cofinite @ nat ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat_le
thf(fact_7173_frequently__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% frequently_cofinite
thf(fact_7174_INFM__iff__infinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% INFM_iff_infinite
thf(fact_7175_not__INFM__eq_I2_J,axiom,
! [A: $tType,A3: A] :
~ ( frequently @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ A3 )
@ ( cofinite @ A ) ) ).
% not_INFM_eq(2)
thf(fact_7176_not__INFM__eq_I1_J,axiom,
! [A: $tType,A3: A] :
~ ( frequently @ A
@ ^ [X3: A] : X3 = A3
@ ( cofinite @ A ) ) ).
% not_INFM_eq(1)
thf(fact_7177_INFM__E,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ~ ! [X2: A] :
~ ( P @ X2 ) ) ).
% INFM_E
thf(fact_7178_INFM__EX,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% INFM_EX
thf(fact_7179_INFM__mono,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( frequently @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% INFM_mono
thf(fact_7180_INFM__disj__distrib,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q2 @ X3 ) )
@ ( cofinite @ A ) )
= ( ( frequently @ A @ P @ ( cofinite @ A ) )
| ( frequently @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% INFM_disj_distrib
thf(fact_7181_INFM__imp__distrib,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ ( cofinite @ A ) )
= ( ( eventually @ A @ P @ ( cofinite @ A ) )
=> ( frequently @ A @ Q2 @ ( cofinite @ A ) ) ) ) ).
% INFM_imp_distrib
thf(fact_7182_Alm__all__def,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( ~ ( frequently @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ ( cofinite @ A ) ) ) ) ).
% Alm_all_def
thf(fact_7183_INFM__conjI,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
=> ( ( eventually @ A @ Q2 @ ( cofinite @ A ) )
=> ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q2 @ X3 ) )
@ ( cofinite @ A ) ) ) ) ).
% INFM_conjI
thf(fact_7184_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,P: B > A > $o,F5: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( frequently @ B
@ ^ [X3: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( P @ X3 @ Y3 ) )
@ F5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X2 )
@ F5 ) ) ) ) ).
% frequently_bex_finite
thf(fact_7185_frequently__bex__finite__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > A > $o,F5: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( frequently @ B
@ ^ [X3: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A4 )
& ( P @ X3 @ Y3 ) )
@ F5 )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X3 )
@ F5 ) ) ) ) ) ).
% frequently_bex_finite_distrib
thf(fact_7186_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X3: A] :
( C3
& ( P @ X3 ) )
@ F5 )
= ( C3
& ( frequently @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(2)
thf(fact_7187_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
& C3 )
@ F5 )
= ( ( frequently @ A @ P @ F5 )
& C3 ) ) ).
% eventually_frequently_const_simps(1)
thf(fact_7188_frequently__ex,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( frequently @ A @ P @ F5 )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% frequently_ex
thf(fact_7189_frequently__disj,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( frequently @ A @ P @ F5 )
=> ( ( frequently @ A @ Q2 @ F5 )
=> ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q2 @ X3 ) )
@ F5 ) ) ) ).
% frequently_disj
thf(fact_7190_frequently__elim1,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( frequently @ A @ P @ F5 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( Q2 @ I2 ) )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_elim1
thf(fact_7191_frequently__disj__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q2 @ X3 ) )
@ F5 )
= ( ( frequently @ A @ P @ F5 )
| ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_disj_iff
thf(fact_7192_not__frequently__False,axiom,
! [A: $tType,F5: filter @ A] :
~ ( frequently @ A
@ ^ [X3: A] : $false
@ F5 ) ).
% not_frequently_False
thf(fact_7193_frequently__imp__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_imp_iff
thf(fact_7194_frequently__rev__mp,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( frequently @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_rev_mp
thf(fact_7195_not__frequently,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( ~ ( frequently @ A @ P @ F5 ) )
= ( eventually @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ F5 ) ) ).
% not_frequently
thf(fact_7196_not__eventually,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( ~ ( eventually @ A @ P @ F5 ) )
= ( frequently @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ F5 ) ) ).
% not_eventually
thf(fact_7197_frequently__def,axiom,
! [A: $tType] :
( ( frequently @ A )
= ( ^ [P2: A > $o,F7: filter @ A] :
~ ( eventually @ A
@ ^ [X3: A] :
~ ( P2 @ X3 )
@ F7 ) ) ) ).
% frequently_def
thf(fact_7198_frequently__mp,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
@ F5 )
=> ( ( frequently @ A @ P @ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_mp
thf(fact_7199_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> C3 )
@ F5 )
= ( ( frequently @ A @ P @ F5 )
=> C3 ) ) ).
% eventually_frequently_const_simps(5)
thf(fact_7200_frequently__all,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X3: A] :
! [X4: B] : ( P @ X3 @ X4 )
@ F5 )
= ( ! [Y9: A > B] :
( frequently @ A
@ ^ [X3: A] : ( P @ X3 @ ( Y9 @ X3 ) )
@ F5 ) ) ) ).
% frequently_all
thf(fact_7201_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ~ ( P @ X3 )
| ( Q2 @ X3 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ) ).
% eventually_frequentlyE
thf(fact_7202_eventually__frequently,axiom,
! [A: $tType,F5: filter @ A,P: A > $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F5 )
=> ( frequently @ A @ P @ F5 ) ) ) ).
% eventually_frequently
thf(fact_7203_frequently__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X3: A] : P
@ F5 )
= ( P
& ( F5
!= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% frequently_const_iff
thf(fact_7204_dsum__Sigma,axiom,
! [B: $tType,A: $tType,A4: set @ ( set @ ( old_node @ A @ B ) ),B3: set @ ( set @ ( old_node @ A @ B ) ),C3: set @ ( set @ ( old_node @ A @ B ) ),D4: set @ ( set @ ( old_node @ A @ B ) )] :
( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( old_dsum @ A @ B
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A4
@ ^ [Uu: set @ ( old_node @ A @ B )] : B3 )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C3
@ ^ [Uu: set @ ( old_node @ A @ B )] : D4 ) )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_usum @ A @ B @ A4 @ C3 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_usum @ A @ B @ B3 @ D4 ) ) ) ).
% dsum_Sigma
thf(fact_7205_dsum__subset__Sigma,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),A4: set @ ( set @ ( old_node @ A @ B ) ),B3: set @ ( set @ ( old_node @ A @ B ) ),S3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),C3: set @ ( set @ ( old_node @ A @ B ) ),D4: set @ ( set @ ( old_node @ A @ B ) )] :
( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ R3
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A4
@ ^ [Uu: set @ ( old_node @ A @ B )] : B3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ S3
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C3
@ ^ [Uu: set @ ( old_node @ A @ B )] : D4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ ( old_dsum @ A @ B @ R3 @ S3 )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_usum @ A @ B @ A4 @ C3 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_usum @ A @ B @ B3 @ D4 ) ) ) ) ) ).
% dsum_subset_Sigma
thf(fact_7206_INFM__inj,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: A > B] :
( ( frequently @ A
@ ^ [X3: A] : ( P @ ( F2 @ X3 ) )
@ ( cofinite @ A ) )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( frequently @ B @ P @ ( cofinite @ B ) ) ) ) ).
% INFM_inj
thf(fact_7207_INFM__finite__Bex__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( frequently @ B
@ ^ [Y3: B] :
? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 @ Y3 ) )
@ ( cofinite @ B ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( frequently @ B @ ( P @ X3 ) @ ( cofinite @ B ) ) ) ) ) ) ).
% INFM_finite_Bex_distrib
thf(fact_7208_INFM__nat__inductI,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [I2: nat] :
( ( P @ I2 )
=> ? [J7: nat] :
( ( ord_less @ nat @ I2 @ J7 )
& ( P @ J7 )
& ( Q2 @ J7 ) ) )
=> ( frequently @ nat @ Q2 @ ( cofinite @ nat ) ) ) ) ).
% INFM_nat_inductI
thf(fact_7209_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_7210_irreflp__irrefl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A5: A,B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R ) )
= ( irrefl @ A @ R ) ) ).
% irreflp_irrefl_eq
thf(fact_7211_subset__mset_Oirreflp__greater,axiom,
! [A: $tType] :
( irreflp @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.irreflp_greater
thf(fact_7212_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% irreflp_greater
thf(fact_7213_dprod__subset__Sigma2,axiom,
! [B: $tType,A: $tType,A4: set @ ( set @ ( old_node @ A @ B ) ),B3: ( set @ ( old_node @ A @ B ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ),C3: set @ ( set @ ( old_node @ A @ B ) ),D4: ( set @ ( old_node @ A @ B ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) )] :
( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ ( old_dprod @ A @ B @ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A4 @ B3 ) @ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C3 @ D4 ) )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_uprod @ A @ B @ A4 @ C3 )
@ ( old_Split @ A @ B @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [X3: set @ ( old_node @ A @ B ),Y3: set @ ( old_node @ A @ B )] : ( old_uprod @ A @ B @ ( B3 @ X3 ) @ ( D4 @ Y3 ) ) ) ) ) ).
% dprod_subset_Sigma2
thf(fact_7214_dprod__subset__Sigma,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),A4: set @ ( set @ ( old_node @ A @ B ) ),B3: set @ ( set @ ( old_node @ A @ B ) ),S3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),C3: set @ ( set @ ( old_node @ A @ B ) ),D4: set @ ( set @ ( old_node @ A @ B ) )] :
( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ R3
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A4
@ ^ [Uu: set @ ( old_node @ A @ B )] : B3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ S3
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C3
@ ^ [Uu: set @ ( old_node @ A @ B )] : D4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ ( old_dprod @ A @ B @ R3 @ S3 )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_uprod @ A @ B @ A4 @ C3 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_uprod @ A @ B @ B3 @ D4 ) ) ) ) ) ).
% dprod_subset_Sigma
thf(fact_7215_uprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_uprod @ A @ B )
= ( ^ [A6: set @ ( set @ ( old_node @ A @ B ) ),B5: set @ ( set @ ( old_node @ A @ B ) )] :
( complete_Sup_Sup @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [X3: set @ ( old_node @ A @ B )] :
( complete_Sup_Sup @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [Y3: set @ ( old_node @ A @ B )] : ( insert2 @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( set @ ( old_node @ A @ B ) ) ) ) )
@ B5 ) )
@ A6 ) ) ) ) ).
% uprod_def
thf(fact_7216_dprod__Sigma,axiom,
! [B: $tType,A: $tType,A4: set @ ( set @ ( old_node @ A @ B ) ),B3: set @ ( set @ ( old_node @ A @ B ) ),C3: set @ ( set @ ( old_node @ A @ B ) ),D4: set @ ( set @ ( old_node @ A @ B ) )] :
( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( old_dprod @ A @ B
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A4
@ ^ [Uu: set @ ( old_node @ A @ B )] : B3 )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C3
@ ^ [Uu: set @ ( old_node @ A @ B )] : D4 ) )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_uprod @ A @ B @ A4 @ C3 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_uprod @ A @ B @ B3 @ D4 ) ) ) ).
% dprod_Sigma
thf(fact_7217_irreflp__multp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( transp @ A @ R3 )
=> ( ( irreflp @ A @ R3 )
=> ( irreflp @ ( multiset @ A ) @ ( multp @ A @ R3 ) ) ) ) ).
% irreflp_multp
thf(fact_7218_multeqp__code__eq__reflclp__multp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( irreflp @ A @ R3 )
=> ( ( transp @ A @ R3 )
=> ( ( multeqp_code @ A @ R3 )
= ( sup_sup @ ( ( multiset @ A ) > ( multiset @ A ) > $o ) @ ( multp @ A @ R3 )
@ ^ [Y5: multiset @ A,Z4: multiset @ A] : Y5 = Z4 ) ) ) ) ).
% multeqp_code_eq_reflclp_multp
thf(fact_7219_transp__trans,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R4: A > A > $o] : ( trans @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ).
% transp_trans
thf(fact_7220_transp__trans__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
= ( trans @ A @ R3 ) ) ).
% transp_trans_eq
thf(fact_7221_transp__inf,axiom,
! [A: $tType,R3: A > A > $o,S3: A > A > $o] :
( ( transp @ A @ R3 )
=> ( ( transp @ A @ S3 )
=> ( transp @ A @ ( inf_inf @ ( A > A > $o ) @ R3 @ S3 ) ) ) ) ).
% transp_inf
thf(fact_7222_subset__mset_Otransp__gr,axiom,
! [A: $tType] :
( transp @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.transp_gr
thf(fact_7223_transp__singleton,axiom,
! [A: $tType,A3: A] :
( transp @ A
@ ^ [X3: A,Y3: A] :
( ( X3 = A3 )
& ( Y3 = A3 ) ) ) ).
% transp_singleton
thf(fact_7224_transp__empty,axiom,
! [A: $tType] :
( transp @ A
@ ^ [X3: A,Y3: A] : $false ) ).
% transp_empty
thf(fact_7225_subset__mset_Otransp__ge,axiom,
! [A: $tType] :
( transp @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.transp_ge
thf(fact_7226_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% transp_gr
thf(fact_7227_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% transp_ge
thf(fact_7228_multp__cancel__add__mset,axiom,
! [A: $tType,R3: A > A > $o,Uu3: A,X7: multiset @ A,Y4: multiset @ A] :
( ( transp @ A @ R3 )
=> ( ( irreflp @ A @ R3 )
=> ( ( multp @ A @ R3 @ ( add_mset @ A @ Uu3 @ X7 ) @ ( add_mset @ A @ Uu3 @ Y4 ) )
= ( multp @ A @ R3 @ X7 @ Y4 ) ) ) ) ).
% multp_cancel_add_mset
thf(fact_7229_multp__cancel,axiom,
! [A: $tType,R3: A > A > $o,X7: multiset @ A,Z6: multiset @ A,Y4: multiset @ A] :
( ( transp @ A @ R3 )
=> ( ( irreflp @ A @ R3 )
=> ( ( multp @ A @ R3 @ ( plus_plus @ ( multiset @ A ) @ X7 @ Z6 ) @ ( plus_plus @ ( multiset @ A ) @ Y4 @ Z6 ) )
= ( multp @ A @ R3 @ X7 @ Y4 ) ) ) ) ).
% multp_cancel
thf(fact_7230_multp__cancel__max,axiom,
! [A: $tType,R3: A > A > $o,X7: multiset @ A,Y4: multiset @ A] :
( ( transp @ A @ R3 )
=> ( ( irreflp @ A @ R3 )
=> ( ( multp @ A @ R3 @ X7 @ Y4 )
= ( multp @ A @ R3 @ ( minus_minus @ ( multiset @ A ) @ X7 @ Y4 ) @ ( minus_minus @ ( multiset @ A ) @ Y4 @ X7 ) ) ) ) ) ).
% multp_cancel_max
thf(fact_7231_equivp__equiv,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A4 )
= ( equiv_equivp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ A4 ) ) ) ).
% equivp_equiv
thf(fact_7232_asymp__asym__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asymp @ A
@ ^ [A5: A,B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R ) )
= ( asym @ A @ R ) ) ).
% asymp_asym_eq
thf(fact_7233_Quotient__crel__quotient,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( equiv_equivp @ A @ R )
=> ( T2
= ( ^ [X3: A] :
( ^ [Y5: B,Z4: B] : Y5 = Z4
@ ( Abs @ X3 ) ) ) ) ) ) ).
% Quotient_crel_quotient
thf(fact_7234_UNIV__typedef__to__equivp,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B] :
( ( type_definition @ B @ A @ Rep @ Abs @ ( top_top @ ( set @ A ) ) )
=> ( equiv_equivp @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% UNIV_typedef_to_equivp
thf(fact_7235_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) ).
% asym_iff
thf(fact_7236_asymD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( asym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) ) ) ).
% asymD
thf(fact_7237_asym_Ointros,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A8: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B7 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A8 ) @ R ) )
=> ( asym @ A @ R ) ) ).
% asym.intros
thf(fact_7238_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A5: set @ ( product_prod @ A @ A )] :
? [R2: set @ ( product_prod @ A @ A )] :
( ( A5 = R2 )
& ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) ) ).
% asym.simps
thf(fact_7239_asym_Ocases,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A3 )
=> ! [A15: A,B13: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A15 @ B13 ) @ A3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B13 @ A15 ) @ A3 ) ) ) ).
% asym.cases
thf(fact_7240_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 ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A3 @ B2 ) @ ( lexord @ A @ R ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A3 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_asymmetric
thf(fact_7241_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R3: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R3 ) )
= ( insert2 @ A @ B2 @ ( range2 @ B @ A @ R3 ) ) ) ).
% Range_insert
thf(fact_7242_Shift__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Shift @ A )
= ( ^ [Kl: set @ ( list @ A ),K4: A] :
( collect @ ( list @ A )
@ ^ [Kl2: list @ A] : ( member @ ( list @ A ) @ ( cons @ A @ K4 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_7243_Range__rtrancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( range2 @ A @ A @ ( transitive_rtrancl @ A @ R ) )
= ( top_top @ ( set @ A ) ) ) ).
% Range_rtrancl
thf(fact_7244_Range__empty,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Range_empty
thf(fact_7245_Range__Id,axiom,
! [A: $tType] :
( ( range2 @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Range_Id
thf(fact_7246_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( range2 @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: B] : ( P @ X3 @ Y3 ) ) ) ).
% Range_Collect_case_prod
thf(fact_7247_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R3 ) )
=> ~ ! [A8: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ A3 ) @ R3 ) ) ).
% Range.cases
thf(fact_7248_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R3 ) )
= ( ? [A5: A,B4: B] :
( ( A3 = B4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R3 ) ) ) ) ).
% Range.simps
thf(fact_7249_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ B @ B2 @ ( range2 @ A @ B @ R3 ) ) ) ).
% Range.intros
thf(fact_7250_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B2 @ ( range2 @ B @ A @ R3 ) )
=> ~ ! [A8: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A8 @ B2 ) @ R3 ) ) ).
% RangeE
thf(fact_7251_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range2 @ B @ A @ R3 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A3 ) @ R3 ) ) ) ).
% Range_iff
thf(fact_7252_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( ( range2 @ B @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) )
= ( R3
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Range_empty_iff
thf(fact_7253_Range__Un__eq,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B3: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( range2 @ B @ A @ A4 ) @ ( range2 @ B @ A @ B3 ) ) ) ).
% Range_Un_eq
thf(fact_7254_Range__Union,axiom,
! [A: $tType,B: $tType,S: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range2 @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range2 @ B @ A ) @ S ) ) ) ).
% Range_Union
thf(fact_7255_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: B] : ( member @ B @ X3 @ ( range2 @ A @ B @ R3 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_7256_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).
% Range_def
thf(fact_7257_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B3: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ A4 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( range2 @ B @ A @ A4 ) @ ( range2 @ B @ A @ B3 ) ) ) ).
% Range_Int_subset
thf(fact_7258_wf__UN,axiom,
! [B: $tType,A: $tType,I4: set @ A,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( wf @ B @ ( R3 @ I2 ) ) )
=> ( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ( member @ A @ J2 @ I4 )
=> ( ( ( R3 @ I2 )
!= ( R3 @ J2 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R3 @ I2 ) ) @ ( range2 @ B @ B @ ( R3 @ J2 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ I4 ) ) ) ) ) ).
% wf_UN
thf(fact_7259_dom__ran__disj__comp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ 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
thf(fact_7260_Domain__rtrancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( domain @ A @ A @ ( transitive_rtrancl @ A @ R ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_rtrancl
thf(fact_7261_Domain__empty,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Domain_empty
thf(fact_7262_Domain__Id,axiom,
! [A: $tType] :
( ( domain @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_Id
thf(fact_7263_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
= ( collect @ A
@ ^ [X3: A] :
? [X4: B] : ( P @ X3 @ X4 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_7264_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 ) )
= ( insert2 @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain_insert
thf(fact_7265_Domain__Union,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S ) ) ) ).
% Domain_Union
thf(fact_7266_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B3: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A4 ) @ ( domain @ A @ B @ B3 ) ) ) ).
% Domain_Un_eq
thf(fact_7267_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( ( domain @ A @ B @ R3 )
= ( bot_bot @ ( set @ A ) ) )
= ( R3
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% Domain_empty_iff
thf(fact_7268_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X3: A] :
? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ).
% Domain_unfold
thf(fact_7269_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 ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_7270_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B7: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ R3 ) ) ).
% Domain.cases
thf(fact_7271_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [A5: A,B4: B] :
( ( A3 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_7272_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_7273_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B7: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ R3 ) ) ).
% DomainE
thf(fact_7274_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_7275_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).
% Domain_def
thf(fact_7276_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( domain @ A @ B @ R3 ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_7277_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B3: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( domain @ A @ B @ A4 ) @ ( domain @ A @ B @ B3 ) ) ) ).
% Domain_Int_subset
thf(fact_7278_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 ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R @ X ) ) @ R ) ) ).
% for_in_RI
thf(fact_7279_Field__def,axiom,
! [A: $tType] :
( ( field2 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ A ) @ ( domain @ A @ A @ R4 ) @ ( range2 @ A @ A @ R4 ) ) ) ) ).
% Field_def
thf(fact_7280_wf__no__path,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range2 @ A @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ R ) ) ).
% wf_no_path
thf(fact_7281_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 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range2 @ A @ A @ R ) ) ) ) ) ).
% wf_min
thf(fact_7282_wf__Un,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( wf @ A @ S3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R3 ) @ ( range2 @ A @ A @ S3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) ) ) ) ) ).
% wf_Un
thf(fact_7283_wf__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
=> ( wf @ A @ X2 ) )
=> ( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
=> ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ X2 ) @ ( range2 @ A @ A @ Xa3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( wf @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) ) ) ).
% wf_Union
thf(fact_7284_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 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R ) @ ( domain @ A @ A @ R ) ) ) ) ) ).
% wf_max
thf(fact_7285_projl__def,axiom,
! [A: $tType,B: $tType] :
( ( sum_projl @ A @ B )
= ( sum_case_sum @ A @ A @ B
@ ^ [X12: A] : X12
@ ^ [Uu2: B] : ( undefined @ A ) ) ) ).
% projl_def
thf(fact_7286_projr__def,axiom,
! [A: $tType,B: $tType] :
( ( sum_projr @ A @ B )
= ( sum_case_sum @ A @ B @ B
@ ^ [Uu2: A] : ( undefined @ B )
@ ^ [X23: B] : X23 ) ) ).
% projr_def
thf(fact_7287_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F: A > B,R2: set @ ( product_prod @ A @ A ),X3: A,Y3: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) @ ( F @ Y3 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_7288_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% old.rec_unit_def
thf(fact_7289_cuts__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,R: set @ ( product_prod @ A @ A ),X: A,G2: A > B] :
( ( ( cut @ A @ B @ F2 @ R @ X )
= ( cut @ A @ B @ G2 @ R @ X ) )
= ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R )
=> ( ( F2 @ Y3 )
= ( G2 @ Y3 ) ) ) ) ) ).
% cuts_eq
thf(fact_7290_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A3: A,R: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R )
=> ( ( cut @ A @ B @ F2 @ R @ A3 @ X )
= ( F2 @ X ) ) ) ).
% cut_apply
thf(fact_7291_adm__lemma,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F5: ( A > B ) > A > B] :
( adm_wf @ A @ B @ R
@ ^ [F: A > B,X3: A] : ( F5 @ ( cut @ A @ B @ F @ R @ X3 ) @ X3 ) ) ).
% adm_lemma
thf(fact_7292_Lcm__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A6: set @ A] :
( gcd_Gcd @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( dvd_dvd @ A @ X3 @ B4 ) ) ) ) ) ) ) ).
% Lcm_Gcd
thf(fact_7293_Lcm__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Lcm @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K5 ) )
= ( gcd_Lcm @ int @ K5 ) ) ).
% Lcm_abs_eq
thf(fact_7294_Lcm__int__eq,axiom,
! [N4: set @ nat] :
( ( gcd_Lcm @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N4 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N4 ) ) ) ).
% Lcm_int_eq
thf(fact_7295_Lcm__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Lcm_UNIV
thf(fact_7296_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_7297_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Lcm @ A @ A4 )
= ( one_one @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_7298_Lcm__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Lcm @ nat
@ ( image2 @ int @ nat
@ ^ [K4: int] : ( nat2 @ ( abs_abs @ int @ K4 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Lcm @ int @ K5 ) ) ) ).
% Lcm_nat_abs_eq
thf(fact_7299_Lcm__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ B,F2: B > A,G2: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( dvd_dvd @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Lcm @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( gcd_Lcm @ A @ ( image2 @ B @ A @ G2 @ A4 ) ) ) ) ) ).
% Lcm_mono
thf(fact_7300_adm__wf__def,axiom,
! [B: $tType,A: $tType] :
( ( adm_wf @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
! [F: A > B,G: A > B,X3: A] :
( ! [Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X3 ) @ R2 )
=> ( ( F @ Z5 )
= ( G @ Z5 ) ) )
=> ( ( F7 @ F @ X3 )
= ( F7 @ G @ X3 ) ) ) ) ) ).
% adm_wf_def
thf(fact_7301_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_7302_Gcd__nat__def,axiom,
( ( gcd_Gcd @ nat )
= ( ^ [M9: set @ nat] :
( gcd_Lcm @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ M9 )
=> ( dvd_dvd @ nat @ D5 @ X3 ) ) ) ) ) ) ).
% Gcd_nat_def
thf(fact_7303_Lcm__int__def,axiom,
( ( gcd_Lcm @ int )
= ( ^ [K7: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K7 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_7304_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A6: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A6
@ ( collect @ A
@ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_7305_Gcd__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A )
= ( ^ [A6: set @ A] :
( gcd_Lcm @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( dvd_dvd @ A @ B4 @ X3 ) ) ) ) ) ) ) ).
% Gcd_Lcm
thf(fact_7306_Lcm__coprime_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( finite_card @ A @ A4 )
!= ( zero_zero @ nat ) )
=> ( ! [A8: A,B7: A] :
( ( member @ A @ A8 @ A4 )
=> ( ( member @ A @ B7 @ A4 )
=> ( ( A8 != B7 )
=> ( algebr8660921524188924756oprime @ A @ A8 @ B7 ) ) ) )
=> ( ( gcd_Lcm @ A @ A4 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X3: A] : X3
@ A4 ) ) ) ) ) ) ).
% Lcm_coprime'
thf(fact_7307_Lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A,B7: A] :
( ( member @ A @ A8 @ A4 )
=> ( ( member @ A @ B7 @ A4 )
=> ( ( A8 != B7 )
=> ( algebr8660921524188924756oprime @ A @ A8 @ B7 ) ) ) )
=> ( ( gcd_Lcm @ A @ A4 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X3: A] : X3
@ A4 ) ) ) ) ) ) ) ).
% Lcm_coprime
thf(fact_7308_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_7309_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_7310_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_7311_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_7312_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_7313_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_7314_Lcm__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% Lcm_singleton
thf(fact_7315_Gcd__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% Gcd_singleton
thf(fact_7316_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,D3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
& ( ( normal6383669964737779283malize @ A @ C2 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_7317_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_7318_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% gcd_mult_right
thf(fact_7319_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_7320_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult
thf(fact_7321_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( A3
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_7322_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_7323_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_7324_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_7325_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C2: A,A4: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A4 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A4 ) ) ) ) ) ).
% Gcd_mult
thf(fact_7326_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A4 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A4 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_7327_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A4 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A4 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_7328_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
thf(fact_7329_Lcm__eq__Max__nat,axiom,
! [M4: set @ nat] :
( ( finite_finite2 @ nat @ M4 )
=> ( ( M4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ! [M3: nat,N3: nat] :
( ( member @ nat @ M3 @ M4 )
=> ( ( member @ nat @ N3 @ M4 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N3 ) @ M4 ) ) )
=> ( ( gcd_Lcm @ nat @ M4 )
= ( lattic643756798349783984er_Max @ nat @ M4 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_7330_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
thf(fact_7331_lcm__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_neg1
thf(fact_7332_lcm__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_neg2
thf(fact_7333_lcm__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_lcm @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% lcm_neg_numeral_2
thf(fact_7334_lcm__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% lcm_neg_numeral_1
thf(fact_7335_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: 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
thf(fact_7336_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_left_normalize
thf(fact_7337_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_right_normalize
thf(fact_7338_lcm__mult__gcd,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% lcm_mult_gcd
thf(fact_7339_gcd__mult__lcm,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% gcd_mult_lcm
thf(fact_7340_Lcm__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% Lcm_2
thf(fact_7341_Lcm__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_7342_Lcm__Un,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( gcd_Lcm @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( gcd_lcm @ A @ ( gcd_Lcm @ A @ A4 ) @ ( gcd_Lcm @ A @ B3 ) ) ) ) ).
% Lcm_Un
thf(fact_7343_lcm__gcd__prod,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% lcm_gcd_prod
thf(fact_7344_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_7345_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% lcm_mult_right
thf(fact_7346_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_7347_lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_lcm @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_coprime
thf(fact_7348_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
thf(fact_7349_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit2
thf(fact_7350_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit1
thf(fact_7351_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_7352_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_7353_Lcm__in__lcm__closed__set__nat,axiom,
! [M4: set @ nat] :
( ( finite_finite2 @ nat @ M4 )
=> ( ( M4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ! [M3: nat,N3: nat] :
( ( member @ nat @ M3 @ M4 )
=> ( ( member @ nat @ N3 @ M4 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N3 ) @ M4 ) ) )
=> ( member @ nat @ ( gcd_Lcm @ nat @ M4 ) @ M4 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_7354_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A5: A,B4: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A5 @ B4 ) @ ( gcd_gcd @ A @ A5 @ B4 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_7355_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_set_eq_fold
thf(fact_7356_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
thf(fact_7357_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_7358_Lcm__nat__def,axiom,
( ( gcd_Lcm @ nat )
= ( ^ [M9: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M9 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M9 ) @ ( zero_zero @ nat ) ) ) ) ).
% Lcm_nat_def
thf(fact_7359_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A6: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A6 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_7360_lcm__1__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
& ( ( N
= ( one_one @ int ) )
| ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% lcm_1_iff_int
thf(fact_7361_Lcm__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_fin.empty
thf(fact_7362_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_7363_lcm__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y )
= ( gcd_lcm @ int @ X @ Y ) ) ).
% lcm_neg1_int
thf(fact_7364_lcm__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y ) )
= ( gcd_lcm @ int @ X @ Y ) ) ).
% lcm_neg2_int
thf(fact_7365_lcm__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_lcm @ int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_lcm @ int @ X @ Y ) ) ) ) ) ) ).
% lcm_cases_int
thf(fact_7366_Lcm__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B3: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( gcd_lcm @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) @ ( semiring_gcd_Lcm_fin @ A @ B3 ) ) ) ) ).
% Lcm_fin.union
thf(fact_7367_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( one_one @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_7368_Lcm__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Lcm @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs @ ( one_one @ int ) ) ) ).
% Lcm_int_set_eq_fold
thf(fact_7369_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B2: A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A4 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A4 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_7370_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Lcm_fin.insert_remove
thf(fact_7371_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Lcm_fin.remove
thf(fact_7372_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_fin.set_eq_fold
thf(fact_7373_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
thf(fact_7374_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A4 )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.remove
thf(fact_7375_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.insert_remove
thf(fact_7376_semilattice__neutr__set__def,axiom,
! [A: $tType] :
( ( lattic5652469242046573047tr_set @ A )
= ( semilattice_neutr @ A ) ) ).
% semilattice_neutr_set_def
thf(fact_7377_semilattice__neutr__set_Oaxioms,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( semilattice_neutr @ A @ F2 @ Z2 ) ) ).
% semilattice_neutr_set.axioms
thf(fact_7378_semilattice__neutr__set_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semilattice_neutr @ A @ F2 @ Z2 )
=> ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 ) ) ).
% semilattice_neutr_set.intro
thf(fact_7379_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% semilattice_neutr_set.empty
thf(fact_7380_semilattice__neutr__set_Ounion,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: set @ A,B3: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( F2 @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A4 ) @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ B3 ) ) ) ) ) ) ).
% semilattice_neutr_set.union
thf(fact_7381_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member @ A @ ( F2 @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_neutr_set.closed
thf(fact_7382_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F: A > A > A,X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ Y3
@ ^ [Z5: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z5 )
@ ^ [Aa3: A] : ( some @ A @ ( F @ Z5 @ Aa3 ) )
@ Y3 )
@ X3 ) ) ) ).
% combine_options_def
thf(fact_7383_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% normalize_div
thf(fact_7384_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% unit_factor_simps(2)
thf(fact_7385_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_7386_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
= A3 ) ) ).
% unit_factor_mult_normalize
thf(fact_7387_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= A3 ) ) ).
% normalize_mult_unit_factor
thf(fact_7388_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
thf(fact_7389_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_7390_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_7391_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: 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
thf(fact_7392_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_7393_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_7394_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_7395_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_7396_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_lcm_right
thf(fact_7397_lcm__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A3: A,B2: A] :
( ( times_times @ A @ K @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% lcm_mult_distrib
thf(fact_7398_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_7399_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_gcd_right
thf(fact_7400_gcd__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A3: A,B2: A] :
( ( times_times @ A @ K @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% gcd_mult_distrib
thf(fact_7401_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% unit_factor_mult
thf(fact_7402_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
thf(fact_7403_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N2: nat] :
( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_7404_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
thf(fact_7405_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_7406_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 )
=> ( ( ( times_times @ A @ A3 @ D3 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_7407_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_7408_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_7409_old_Orec__sum__def,axiom,
! [B: $tType,T: $tType,A: $tType] :
( ( sum_rec_sum @ A @ T @ B )
= ( ^ [F12: A > T,F23: B > T,X3: sum_sum @ A @ B] : ( the @ T @ ( sum_rec_set_sum @ A @ T @ B @ F12 @ F23 @ X3 ) ) ) ) ).
% old.rec_sum_def
thf(fact_7410_mk__less__def,axiom,
! [A: $tType] :
( ( partial_mk_less @ A )
= ( ^ [R2: A > A > $o,X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
& ~ ( R2 @ Y3 @ X3 ) ) ) ) ).
% mk_less_def
thf(fact_7411_Random__Pred_Ounion__def,axiom,
! [A: $tType] :
( ( random_union @ A )
= ( ^ [R15: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R25: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S2: 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 ) )
@ ^ [P13: pred @ A,S10: 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 ) )
@ ^ [P25: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P13 @ P25 ) )
@ ( R25 @ S10 ) )
@ ( R15 @ S2 ) ) ) ) ).
% Random_Pred.union_def
thf(fact_7412_img__ord__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( partial_img_ord @ A @ C @ B )
= ( ^ [F: A > C,Ord2: C > C > B,X3: A,Y3: A] : ( Ord2 @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) ).
% img_ord_def
thf(fact_7413_Random__Pred_Oempty__def,axiom,
! [A: $tType] :
( ( random_empty @ A )
= ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% Random_Pred.empty_def
thf(fact_7414_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( random_bind @ A @ B )
= ( ^ [R2: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),F: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S2: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P2: pred @ A,S10: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [S1: product_prod @ code_natural @ code_natural] :
( product_Pair @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural )
@ ( bind2 @ A @ B @ P2
@ ^ [A5: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F @ A5 @ S1 ) ) ) )
@ ( split_seed @ S10 ) )
@ ( R2 @ S2 ) ) ) ) ).
% Random_Pred.bind_def
thf(fact_7415_sup__bind,axiom,
! [A: $tType,B: $tType,P: pred @ B,Q2: pred @ B,R: B > ( pred @ A )] :
( ( bind2 @ B @ A @ ( sup_sup @ ( pred @ B ) @ P @ Q2 ) @ R )
= ( sup_sup @ ( pred @ A ) @ ( bind2 @ B @ A @ P @ R ) @ ( bind2 @ B @ A @ Q2 @ R ) ) ) ).
% sup_bind
thf(fact_7416_Sup__bind,axiom,
! [A: $tType,B: $tType,A4: set @ ( pred @ B ),F2: B > ( pred @ A )] :
( ( bind2 @ B @ A @ ( complete_Sup_Sup @ ( pred @ B ) @ A4 ) @ F2 )
= ( complete_Sup_Sup @ ( pred @ A )
@ ( image2 @ ( pred @ B ) @ ( pred @ A )
@ ^ [X3: pred @ B] : ( bind2 @ B @ A @ X3 @ F2 )
@ A4 ) ) ) ).
% Sup_bind
thf(fact_7417_Predicate_Obind__bind,axiom,
! [B: $tType,A: $tType,C: $tType,P: pred @ C,Q2: C > ( pred @ B ),R: B > ( pred @ A )] :
( ( bind2 @ B @ A @ ( bind2 @ C @ B @ P @ Q2 ) @ R )
= ( bind2 @ C @ A @ P
@ ^ [X3: C] : ( bind2 @ B @ A @ ( Q2 @ X3 ) @ R ) ) ) ).
% Predicate.bind_bind
thf(fact_7418_bottom__bind,axiom,
! [B: $tType,A: $tType,P: B > ( pred @ A )] :
( ( bind2 @ B @ A @ ( bot_bot @ ( pred @ B ) ) @ P )
= ( bot_bot @ ( pred @ A ) ) ) ).
% bottom_bind
thf(fact_7419_singleton__sup,axiom,
! [A: $tType,A4: pred @ A,Default: product_unit > A,B3: pred @ A] :
( ( ( A4
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( singleton @ A @ Default @ B3 ) ) )
& ( ( A4
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( ( B3
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( singleton @ A @ Default @ A4 ) ) )
& ( ( B3
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( ( ( singleton @ A @ Default @ A4 )
= ( singleton @ A @ Default @ B3 ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( singleton @ A @ Default @ A4 ) ) )
& ( ( ( singleton @ A @ Default @ A4 )
!= ( singleton @ A @ Default @ B3 ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( Default @ product_Unity ) ) ) ) ) ) ) ) ).
% singleton_sup
thf(fact_7420_singleton__bot,axiom,
! [A: $tType,Default: product_unit > A] :
( ( singleton @ A @ Default @ ( bot_bot @ ( pred @ A ) ) )
= ( Default @ product_Unity ) ) ).
% singleton_bot
thf(fact_7421_subset_Osuc__Union__closed__empty,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) ) ) ).
% subset.suc_Union_closed_empty
thf(fact_7422_pred__on_Osuc__Union__closed__empty,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ A @ A4 @ P ) ) ).
% pred_on.suc_Union_closed_empty
thf(fact_7423_pred__on_Osuc__Union__closed__def,axiom,
! [A: $tType] :
( ( pred_s596693808085603175closed @ A )
= ( ^ [A6: set @ A,P2: A > A > $o] : ( collect @ ( set @ A ) @ ( pred_s7749564232668923593losedp @ A @ A6 @ P2 ) ) ) ) ).
% pred_on.suc_Union_closed_def
thf(fact_7424_subset_Osuc__Union__closed__def,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( pred_s596693808085603175closed @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) )
= ( collect @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% subset.suc_Union_closed_def
thf(fact_7425_eval__bind,axiom,
! [A: $tType,B: $tType,P: pred @ B,F2: B > ( pred @ A )] :
( ( eval @ A @ ( bind2 @ B @ A @ P @ F2 ) )
= ( eval @ A @ ( complete_Sup_Sup @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F2 @ ( collect @ B @ ( eval @ B @ P ) ) ) ) ) ) ).
% eval_bind
thf(fact_7426_singleton__sup__aux,axiom,
! [A: $tType,A4: pred @ A,Default: product_unit > A,B3: pred @ A] :
( ( ( A4
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( singleton @ A @ Default @ B3 ) ) )
& ( ( A4
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( ( B3
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( singleton @ A @ Default @ A4 ) ) )
& ( ( B3
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ ( single @ A @ ( singleton @ A @ Default @ A4 ) ) @ ( single @ A @ ( singleton @ A @ Default @ B3 ) ) ) ) ) ) ) ) ) ).
% singleton_sup_aux
thf(fact_7427_eval__inf,axiom,
! [A: $tType,P: pred @ A,Q2: pred @ A] :
( ( eval @ A @ ( inf_inf @ ( pred @ A ) @ P @ Q2 ) )
= ( inf_inf @ ( A > $o ) @ ( eval @ A @ P ) @ ( eval @ A @ Q2 ) ) ) ).
% eval_inf
thf(fact_7428_eval__top,axiom,
! [A: $tType] :
( ( eval @ A @ ( top_top @ ( pred @ A ) ) )
= ( top_top @ ( A > $o ) ) ) ).
% eval_top
thf(fact_7429_eval__compl,axiom,
! [A: $tType,P: pred @ A] :
( ( eval @ A @ ( uminus_uminus @ ( pred @ A ) @ P ) )
= ( uminus_uminus @ ( A > $o ) @ ( eval @ A @ P ) ) ) ).
% eval_compl
thf(fact_7430_eval__bot,axiom,
! [A: $tType] :
( ( eval @ A @ ( bot_bot @ ( pred @ A ) ) )
= ( bot_bot @ ( A > $o ) ) ) ).
% eval_bot
thf(fact_7431_eval__sup,axiom,
! [A: $tType,P: pred @ A,Q2: pred @ A] :
( ( eval @ A @ ( sup_sup @ ( pred @ A ) @ P @ Q2 ) )
= ( sup_sup @ ( A > $o ) @ ( eval @ A @ P ) @ ( eval @ A @ Q2 ) ) ) ).
% eval_sup
thf(fact_7432_supE,axiom,
! [A: $tType,A4: pred @ A,B3: pred @ A,X: A] :
( ( eval @ A @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) @ X )
=> ( ~ ( eval @ A @ A4 @ X )
=> ( eval @ A @ B3 @ X ) ) ) ).
% supE
thf(fact_7433_supI1,axiom,
! [A: $tType,A4: pred @ A,X: A,B3: pred @ A] :
( ( eval @ A @ A4 @ X )
=> ( eval @ A @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) @ X ) ) ).
% supI1
thf(fact_7434_supI2,axiom,
! [A: $tType,B3: pred @ A,X: A,A4: pred @ A] :
( ( eval @ A @ B3 @ X )
=> ( eval @ A @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) @ X ) ) ).
% supI2
thf(fact_7435_botE,axiom,
! [A: $tType,X: A] :
~ ( eval @ A @ ( bot_bot @ ( pred @ A ) ) @ X ) ).
% botE
thf(fact_7436_not__bot,axiom,
! [A: $tType,A4: pred @ A] :
( ( A4
!= ( bot_bot @ ( pred @ A ) ) )
=> ~ ! [X2: A] :
~ ( eval @ A @ A4 @ X2 ) ) ).
% not_bot
thf(fact_7437_single__not__bot,axiom,
! [A: $tType,X: A] :
( ( single @ A @ X )
!= ( bot_bot @ ( pred @ A ) ) ) ).
% single_not_bot
thf(fact_7438_unit__pred__cases,axiom,
! [P: ( pred @ product_unit ) > $o,Q2: pred @ product_unit] :
( ( P @ ( bot_bot @ ( pred @ product_unit ) ) )
=> ( ( P @ ( single @ product_unit @ product_Unity ) )
=> ( P @ Q2 ) ) ) ).
% unit_pred_cases
thf(fact_7439_Predicate_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind2 @ A @ B )
= ( ^ [P2: pred @ A,F: A > ( pred @ B )] : ( complete_Sup_Sup @ ( pred @ B ) @ ( image2 @ A @ ( pred @ B ) @ F @ ( collect @ A @ ( eval @ A @ P2 ) ) ) ) ) ) ).
% Predicate.bind_def
thf(fact_7440_singleton__sup__single__single,axiom,
! [A: $tType,X: A,Y: A,Default: product_unit > A] :
( ( ( X = Y )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ ( single @ A @ X ) @ ( single @ A @ Y ) ) )
= X ) )
& ( ( X != Y )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ ( single @ A @ X ) @ ( single @ A @ Y ) ) )
= ( Default @ product_Unity ) ) ) ) ).
% singleton_sup_single_single
thf(fact_7441_Random__Pred_ORandom__def,axiom,
! [A: $tType] :
( ( random_Random @ A )
= ( ^ [G: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) )] : ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) @ G @ ( comp @ ( pred @ A ) @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( comp @ A @ ( pred @ A ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( single @ A ) @ ( product_fst @ A @ ( product_unit > code_term ) ) ) ) ) ) ) ).
% Random_Pred.Random_def
thf(fact_7442_pred__of__set__fold__sup,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pred_of_set @ A @ A4 )
= ( finite_fold @ ( pred @ A ) @ ( pred @ A ) @ ( sup_sup @ ( pred @ A ) ) @ ( bot_bot @ ( pred @ A ) ) @ ( image2 @ A @ ( pred @ A ) @ ( single @ A ) @ A4 ) ) ) ) ).
% pred_of_set_fold_sup
thf(fact_7443_pred__of__set__set__fold__sup,axiom,
! [A: $tType,Xs: list @ A] :
( ( pred_of_set @ A @ ( set2 @ A @ Xs ) )
= ( 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
thf(fact_7444_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
thf(fact_7445_Random__Pred_Onot__randompred__def,axiom,
( random6974930770145893639ompred
= ( ^ [P2: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ),S2: 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 ) )
@ ^ [P14: pred @ product_unit,S10: product_prod @ code_natural @ code_natural] : ( if @ ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( eval @ product_unit @ P14 @ product_Unity ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ product_unit ) ) @ S10 ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ product_unit @ product_Unity ) @ S10 ) )
@ ( P2 @ S2 ) ) ) ) ).
% Random_Pred.not_randompred_def
thf(fact_7446_Random__Pred_Osingle__def,axiom,
! [A: $tType] :
( ( random_single @ A )
= ( ^ [X3: A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ A @ X3 ) ) ) ) ).
% Random_Pred.single_def
thf(fact_7447_not__pred__eq,axiom,
( not_pred
= ( ^ [P2: pred @ product_unit] : ( if @ ( pred @ product_unit ) @ ( eval @ product_unit @ P2 @ product_Unity ) @ ( bot_bot @ ( pred @ product_unit ) ) @ ( single @ product_unit @ product_Unity ) ) ) ) ).
% not_pred_eq
thf(fact_7448_if__pred__eq,axiom,
( if_pred
= ( ^ [B4: $o] : ( if @ ( pred @ product_unit ) @ B4 @ ( single @ product_unit @ product_Unity ) @ ( bot_bot @ ( pred @ product_unit ) ) ) ) ) ).
% if_pred_eq
thf(fact_7449_eval__map,axiom,
! [A: $tType,B: $tType,F2: B > A,P: pred @ B] :
( ( eval @ A @ ( map2 @ B @ A @ F2 @ P ) )
= ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [X3: B] :
( ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( F2 @ X3 ) )
@ ( collect @ B @ ( eval @ B @ P ) ) ) ) ) ).
% eval_map
thf(fact_7450_the__eqI,axiom,
! [A: $tType,P: pred @ A,X: A] :
( ( ( the @ A @ ( eval @ A @ P ) )
= X )
=> ( ( the3 @ A @ P )
= X ) ) ).
% the_eqI
thf(fact_7451_Predicate_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map2 @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( pred @ A ) ) ) ).
% Predicate.map.identity
thf(fact_7452_Predicate_Othe__def,axiom,
! [A: $tType] :
( ( the3 @ A )
= ( ^ [A6: pred @ A] : ( the @ A @ ( eval @ A @ A6 ) ) ) ) ).
% Predicate.the_def
thf(fact_7453_the__eq,axiom,
! [A: $tType] :
( ( the3 @ A )
= ( ^ [A6: pred @ A] :
( singleton @ A
@ ^ [X3: product_unit] :
( abort @ A @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu: product_unit] : ( the3 @ A @ A6 ) )
@ A6 ) ) ) ).
% the_eq
thf(fact_7454_uminus__pred__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( pred @ A ) )
= ( ^ [P2: pred @ A] : ( pred3 @ A @ ( uminus_uminus @ ( A > $o ) @ ( eval @ A @ P2 ) ) ) ) ) ).
% uminus_pred_def
thf(fact_7455_pred__of__set__def,axiom,
! [A: $tType] :
( ( pred_of_set @ A )
= ( comp @ ( A > $o ) @ ( pred @ A ) @ ( set @ A ) @ ( pred3 @ A )
@ ^ [A6: set @ A,X3: A] : ( member @ A @ X3 @ A6 ) ) ) ).
% pred_of_set_def
thf(fact_7456_bot__pred__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( pred @ A ) )
= ( pred3 @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_pred_def
thf(fact_7457_not__predE,axiom,
! [P: $o,X: product_unit] :
( ( eval @ product_unit
@ ( not_pred
@ ( pred3 @ product_unit
@ ^ [U2: product_unit] : P ) )
@ X )
=> ~ P ) ).
% not_predE
thf(fact_7458_top__pred__def,axiom,
! [A: $tType] :
( ( top_top @ ( pred @ A ) )
= ( pred3 @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_pred_def
thf(fact_7459_sup__pred__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( pred @ A ) )
= ( ^ [P2: pred @ A,Q: pred @ A] : ( pred3 @ A @ ( sup_sup @ ( A > $o ) @ ( eval @ A @ P2 ) @ ( eval @ A @ Q ) ) ) ) ) ).
% sup_pred_def
thf(fact_7460_not__predI,axiom,
! [P: $o] :
( ~ P
=> ( eval @ product_unit
@ ( not_pred
@ ( pred3 @ product_unit
@ ^ [U2: product_unit] : P ) )
@ product_Unity ) ) ).
% not_predI
thf(fact_7461_inf__pred__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( pred @ A ) )
= ( ^ [P2: pred @ A,Q: pred @ A] : ( pred3 @ A @ ( inf_inf @ ( A > $o ) @ ( eval @ A @ P2 ) @ ( eval @ A @ Q ) ) ) ) ) ).
% inf_pred_def
thf(fact_7462_contains__pred__def,axiom,
! [A: $tType] :
( ( predic7144156976422707464s_pred @ A )
= ( ^ [A6: set @ A,X3: A] : ( if @ ( pred @ product_unit ) @ ( member @ A @ X3 @ A6 ) @ ( single @ product_unit @ product_Unity ) @ ( bot_bot @ ( pred @ product_unit ) ) ) ) ) ).
% contains_pred_def
thf(fact_7463_singleton__def,axiom,
! [A: $tType] :
( ( singleton @ A )
= ( ^ [Default2: product_unit > A,A6: pred @ A] :
( if @ A
@ ? [X3: A] :
( ( eval @ A @ A6 @ X3 )
& ! [Y3: A] :
( ( eval @ A @ A6 @ Y3 )
=> ( Y3 = X3 ) ) )
@ ( the @ A @ ( eval @ A @ A6 ) )
@ ( Default2 @ product_Unity ) ) ) ) ).
% singleton_def
thf(fact_7464_Nitpick_OEx1__unfold,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P2: A > $o] :
? [X3: A] :
( ( collect @ A @ P2 )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Nitpick.Ex1_unfold
thf(fact_7465_Predicate__Compile_Ocontains__pred__eq,axiom,
! [A: $tType] :
( ( predic7144156976422707464s_pred @ A )
= ( ^ [A6: set @ A,X3: A] :
( pred3 @ product_unit
@ ^ [Y3: product_unit] : ( predicate_contains @ A @ A6 @ X3 ) ) ) ) ).
% Predicate_Compile.contains_pred_eq
thf(fact_7466_THE__default__def,axiom,
! [A: $tType] :
( ( fun_THE_default @ A )
= ( ^ [D5: A,P2: A > $o] :
( if @ A
@ ? [X3: A] :
( ( P2 @ X3 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X3 ) ) )
@ ( the @ A @ P2 )
@ D5 ) ) ) ).
% THE_default_def
thf(fact_7467_fundef__ex1__uniqueness,axiom,
! [B: $tType,A: $tType,F2: A > B,D3: A > B,G5: A > B > $o,X: A,H3: A > B] :
( ( F2
= ( ^ [X3: A] : ( fun_THE_default @ B @ ( D3 @ X3 ) @ ( G5 @ X3 ) ) ) )
=> ( ? [X5: B] :
( ( G5 @ X @ X5 )
& ! [Y2: B] :
( ( G5 @ X @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( ( G5 @ X @ ( H3 @ X ) )
=> ( ( H3 @ X )
= ( F2 @ X ) ) ) ) ) ).
% fundef_ex1_uniqueness
thf(fact_7468_fundef__ex1__existence,axiom,
! [B: $tType,A: $tType,F2: A > B,D3: A > B,G5: A > B > $o,X: A] :
( ( F2
= ( ^ [X3: A] : ( fun_THE_default @ B @ ( D3 @ X3 ) @ ( G5 @ X3 ) ) ) )
=> ( ? [X5: B] :
( ( G5 @ X @ X5 )
& ! [Y2: B] :
( ( G5 @ X @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( G5 @ X @ ( F2 @ X ) ) ) ) ).
% fundef_ex1_existence
thf(fact_7469_fundef__default__value,axiom,
! [B: $tType,A: $tType,F2: A > B,D3: A > B,G5: A > B > $o,D4: A > $o,X: A] :
( ( F2
= ( ^ [X3: A] : ( fun_THE_default @ B @ ( D3 @ X3 ) @ ( G5 @ X3 ) ) ) )
=> ( ! [X2: A,Y2: B] :
( ( G5 @ X2 @ Y2 )
=> ( D4 @ X2 ) )
=> ( ~ ( D4 @ X )
=> ( ( F2 @ X )
= ( D3 @ X ) ) ) ) ) ).
% fundef_default_value
thf(fact_7470_fundef__ex1__iff,axiom,
! [A: $tType,B: $tType,F2: A > B,D3: A > B,G5: A > B > $o,X: A,Y: B] :
( ( F2
= ( ^ [X3: A] : ( fun_THE_default @ B @ ( D3 @ X3 ) @ ( G5 @ X3 ) ) ) )
=> ( ? [X5: B] :
( ( G5 @ X @ X5 )
& ! [Y2: B] :
( ( G5 @ X @ Y2 )
=> ( Y2 = X5 ) ) )
=> ( ( G5 @ X @ Y )
= ( ( F2 @ X )
= Y ) ) ) ) ).
% fundef_ex1_iff
thf(fact_7471_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X2: A] :
( ( A3
= ( uminus_uminus @ A @ X2 ) )
=> ~ ( neg_numeral_is_num @ A @ X2 ) )
=> ~ ! [X2: A,Y2: A] :
( ( A3
= ( plus_plus @ A @ X2 @ Y2 ) )
=> ( ( neg_numeral_is_num @ A @ X2 )
=> ~ ( neg_numeral_is_num @ A @ Y2 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_7472_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A5: A] :
( ( A5
= ( one_one @ A ) )
| ? [X3: A] :
( ( A5
= ( uminus_uminus @ A @ X3 ) )
& ( neg_numeral_is_num @ A @ X3 ) )
| ? [X3: A,Y3: A] :
( ( A5
= ( plus_plus @ A @ X3 @ Y3 ) )
& ( neg_numeral_is_num @ A @ X3 )
& ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_7473_is__num__normalize_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% is_num_normalize(5)
thf(fact_7474_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_7475_open__typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( equiv_part_equivp @ B @ ( bNF_eq_onp @ B @ P ) ) ) ).
% open_typedef_to_part_equivp
thf(fact_7476_typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: set @ B] :
( ( type_definition @ A @ B @ Rep @ Abs @ S )
=> ( equiv_part_equivp @ B
@ ( bNF_eq_onp @ B
@ ^ [X3: B] : ( member @ B @ X3 @ S ) ) ) ) ).
% typedef_to_part_equivp
thf(fact_7477_part__equivp__typedef,axiom,
! [A: $tType,R: A > A > $o] :
( ( equiv_part_equivp @ A @ R )
=> ? [D2: set @ A] :
( member @ ( set @ A ) @ D2
@ ( collect @ ( set @ A )
@ ^ [C5: set @ A] :
? [X3: A] :
( ( R @ X3 @ X3 )
& ( C5
= ( collect @ A @ ( R @ X3 ) ) ) ) ) ) ) ).
% part_equivp_typedef
thf(fact_7478_snds__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_snds @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( collect @ B @ ( basic_sndsp @ A @ B @ P6 ) ) ) ) ).
% snds_def
thf(fact_7479_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_7480_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_7481_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_7482_setr__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_setr @ A @ B )
= ( ^ [S2: sum_sum @ A @ B] : ( collect @ B @ ( basic_setrp @ A @ B @ S2 ) ) ) ) ).
% setr_def
thf(fact_7483_setl__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_setl @ A @ B )
= ( ^ [S2: sum_sum @ A @ B] : ( collect @ A @ ( basic_setlp @ A @ B @ S2 ) ) ) ) ).
% setl_def
thf(fact_7484_fsts__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( collect @ A @ ( basic_fstsp @ A @ B @ P6 ) ) ) ) ).
% fsts_def
thf(fact_7485_pick__middlep__def,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( bNF_pick_middlep @ B @ A @ C )
= ( ^ [P2: B > A > $o,Q: A > C > $o,A5: B,C5: C] :
( fChoice @ A
@ ^ [B4: A] :
( ( P2 @ A5 @ B4 )
& ( Q @ B4 @ C5 ) ) ) ) ) ).
% pick_middlep_def
thf(fact_7486_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P2: C > A > $o,Q: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P2 @ Q @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_7487_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P2: A > B > $o,Q: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_7488_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F: A > B,A6: set @ A,X3: B] :
( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A6 )
& ( ( F @ Y3 )
= X3 ) ) ) ) ) ).
% inv_on_def
thf(fact_7489_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
thf(fact_7490_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod'_def
thf(fact_7491_subset__mset_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_order @ ( multiset @ A ) @ ( union_mset @ A )
@ ^ [A6: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ B5 @ A6 )
@ ^ [A6: multiset @ A,B5: multiset @ A] : ( subset_mset @ A @ B5 @ A6 ) ) ).
% subset_mset.semilattice_order_axioms
thf(fact_7492_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_neutr @ A @ F2 @ Z2 )
=> ( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less ) ) ) ).
% semilattice_neutr_order.intro
thf(fact_7493_semilattice__neutr__order__def,axiom,
! [A: $tType] :
( ( semila1105856199041335345_order @ A )
= ( ^ [F: A > A > A,Z5: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_neutr @ A @ F @ Z5 )
& ( semilattice_order @ A @ F @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_neutr_order_def
thf(fact_7494_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( semilattice_order @ A @ F2 @ Less_eq @ Less ) ) ).
% semilattice_neutr_order.axioms(2)
thf(fact_7495_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_7496_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ C2 )
=> ( Less @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_7497_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( A3
= ( F2 @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_7498_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ ( F2 @ B2 @ C2 ) )
=> ~ ( ( Less @ A3 @ B2 )
=> ~ ( Less @ A3 @ C2 ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_7499_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_7500_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C2 )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_7501_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F2 @ B2 @ C2 ) )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ A3 @ C2 ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_7502_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
= ( ( F2 @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_7503_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( F2 @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_7504_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ B2 ) ) ).
% semilattice_order.cobounded2
thf(fact_7505_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ A3 ) ) ).
% semilattice_order.cobounded1
thf(fact_7506_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( A3
= ( F2 @ A3 @ B2 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_7507_semilattice__order_OboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ A3 @ C2 )
=> ( Less_eq @ A3 @ ( F2 @ B2 @ C2 ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_7508_semilattice__order_OboundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F2 @ B2 @ C2 ) )
=> ~ ( ( Less_eq @ A3 @ B2 )
=> ~ ( Less_eq @ A3 @ C2 ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_7509_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ B2 @ A3 )
=> ( ( F2 @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb4
thf(fact_7510_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( F2 @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb3
thf(fact_7511_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
=> ( ( F2 @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb2
thf(fact_7512_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( F2 @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb1
thf(fact_7513_semilattice__order_OorderI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( A3
= ( F2 @ A3 @ B2 ) )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% semilattice_order.orderI
thf(fact_7514_semilattice__order_OorderE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( A3
= ( F2 @ A3 @ B2 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_7515_semilattice__order_Omono,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A,D3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C2 )
=> ( ( Less_eq @ B2 @ D3 )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ ( F2 @ C2 @ D3 ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_7516_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
thf(fact_7517_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
thf(fact_7518_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_7519_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_7520_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
= N ) ).
% Suc_natural_minus_one
thf(fact_7521_comp__fun__idem__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite_comp_fun_idem @ A @ B )
= ( finite673082921795544331dem_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% comp_fun_idem_def'
thf(fact_7522_gcd__nat_Osemilattice__order__axioms,axiom,
( semilattice_order @ nat @ ( gcd_gcd @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.semilattice_order_axioms
thf(fact_7523_comp__fun__idem__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( finite_comp_fun_idem @ A @ A @ ( inf_inf @ A ) ) ) ).
% comp_fun_idem_inf
thf(fact_7524_comp__fun__idem__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( finite_comp_fun_idem @ A @ A @ ( sup_sup @ A ) ) ) ).
% comp_fun_idem_sup
thf(fact_7525_random__aux__set_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
( ! [J2: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( zero_zero @ code_natural ) @ J2 ) )
=> ~ ! [I2: code_natural,J2: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( code_Suc @ I2 ) @ J2 ) ) ) ).
% random_aux_set.cases
thf(fact_7526_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
thf(fact_7527_comm__monoid__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType,F2: A > A > A,Z2: A,A4: multiset @ B] :
( ( comm_monoid_mset @ A @ F2 @ Z2 )
=> ( ( comm_monoid_F @ A @ F2 @ Z2
@ ( image_mset @ B @ A
@ ^ [Uu: B] : Z2
@ A4 ) )
= Z2 ) ) ).
% comm_monoid_mset.neutral_const
thf(fact_7528_comm__monoid__mset_Odistrib,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,G2: B > A,H3: B > A,A4: multiset @ B] :
( ( comm_monoid_mset @ A @ F2 @ Z2 )
=> ( ( comm_monoid_F @ A @ F2 @ Z2
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( F2 @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A4 ) )
= ( F2 @ ( comm_monoid_F @ A @ F2 @ Z2 @ ( image_mset @ B @ A @ G2 @ A4 ) ) @ ( comm_monoid_F @ A @ F2 @ Z2 @ ( image_mset @ B @ A @ H3 @ A4 ) ) ) ) ) ).
% comm_monoid_mset.distrib
thf(fact_7529_comm__monoid__mset__def,axiom,
! [A: $tType] :
( ( comm_monoid_mset @ A )
= ( comm_monoid @ A ) ) ).
% comm_monoid_mset_def
thf(fact_7530_comm__monoid__mset_Oaxioms,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( comm_monoid_mset @ A @ F2 @ Z2 )
=> ( comm_monoid @ A @ F2 @ Z2 ) ) ).
% comm_monoid_mset.axioms
thf(fact_7531_comm__monoid__mset_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( comm_monoid_mset @ A @ F2 @ Z2 ) ) ).
% comm_monoid_mset.intro
thf(fact_7532_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
thf(fact_7533_comm__monoid__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > A > A,Z2: A,G2: B > C > A,B3: multiset @ C,A4: multiset @ B] :
( ( comm_monoid_mset @ A @ F2 @ Z2 )
=> ( ( comm_monoid_F @ A @ F2 @ Z2
@ ( image_mset @ B @ A
@ ^ [I3: B] : ( comm_monoid_F @ A @ F2 @ Z2 @ ( image_mset @ C @ A @ ( G2 @ I3 ) @ B3 ) )
@ A4 ) )
= ( comm_monoid_F @ A @ F2 @ Z2
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_monoid_F @ A @ F2 @ Z2
@ ( image_mset @ B @ A
@ ^ [I3: B] : ( G2 @ I3 @ J3 )
@ A4 ) )
@ B3 ) ) ) ) ).
% comm_monoid_mset.swap
thf(fact_7534_or__num_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_un6697907153464112080or_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit1 @ N3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ N3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit0 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
thf(fact_7535_bdd__below__primitive__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( condit16957441358409770ng_bdd @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ).
% bdd_below_primitive_def
thf(fact_7536_subset__mset_Obdd__below__primitive__def,axiom,
! [A: $tType] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) )
= ( condit16957441358409770ng_bdd @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 ) ) ) ).
% subset_mset.bdd_below_primitive_def
thf(fact_7537_option_Osimps_I15_J,axiom,
! [A: $tType,X22: A] :
( ( set_option @ A @ ( some @ A @ X22 ) )
= ( insert2 @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_7538_those_Osimps_I2_J,axiom,
! [A: $tType,X: option @ A,Xs: list @ ( option @ A )] :
( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs ) )
= ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
@ ^ [Y3: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( those @ A @ Xs ) )
@ X ) ) ).
% those.simps(2)
thf(fact_7539_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_7540_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_7541_option_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( rel_option @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F2 ) )
= ( bNF_Grp @ ( option @ A ) @ ( option @ B )
@ ( collect @ ( option @ A )
@ ^ [X3: option @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_option @ A @ X3 ) @ A4 ) )
@ ( map_option @ A @ B @ F2 ) ) ) ).
% option.rel_Grp
thf(fact_7542_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R2: A > B > $o,A5: option @ A,B4: option @ B] :
? [Z5: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A5 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B4 ) ) ) ) ).
% option.in_rel
thf(fact_7543_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G2: B > C,Y: option @ B] :
( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G2 @ Y ) )
= ( rel_option @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G2 @ Y3 ) )
@ X
@ Y ) ) ).
% option.rel_map(2)
thf(fact_7544_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: option @ A,Y: option @ B] :
( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X ) @ Y )
= ( rel_option @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% option.rel_map(1)
thf(fact_7545_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [Option2: option @ A] :
( Option2
!= ( none @ A ) )
@ ^ [Option2: option @ B] :
( Option2
!= ( none @ B ) ) ) ).
% option.disc_transfer(2)
thf(fact_7546_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [Option2: option @ A] :
( Option2
= ( none @ A ) )
@ ^ [Option2: option @ B] :
( Option2
= ( none @ B ) ) ) ).
% option.disc_transfer(1)
thf(fact_7547_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R2: A > B > $o,X3: option @ A,Y3: option @ B] :
( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
@ ^ [A5: option @ A,B4: option @ B] :
( case_option @ $o @ A
@ ( case_option @ $o @ B @ $true
@ ^ [C5: B] : $false
@ B4 )
@ ^ [Z5: A] : ( case_option @ $o @ B @ $false @ ( R2 @ Z5 ) @ B4 )
@ A5 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X3 @ Y3 ) ) ) ) ).
% rel_option_iff
thf(fact_7548_rel__option__inf,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B3: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A4 ) @ ( rel_option @ A @ B @ B3 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A4 @ B3 ) ) ) ).
% rel_option_inf
thf(fact_7549_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R2: A > B > $o] :
( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% option.rel_compp_Grp
thf(fact_7550_preordering__bdd_OInt1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: set @ A,B3: set @ A] :
( ( condit622319405099724424ng_bdd @ A @ Less_eq @ Less )
=> ( ( condit16957441358409770ng_bdd @ A @ Less_eq @ A4 )
=> ( condit16957441358409770ng_bdd @ A @ Less_eq @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% preordering_bdd.Int1
thf(fact_7551_preordering__bdd_OInt2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,B3: set @ A,A4: set @ A] :
( ( condit622319405099724424ng_bdd @ A @ Less_eq @ Less )
=> ( ( condit16957441358409770ng_bdd @ A @ Less_eq @ B3 )
=> ( condit16957441358409770ng_bdd @ A @ Less_eq @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% preordering_bdd.Int2
thf(fact_7552_subset__mset_Obdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( condit622319405099724424ng_bdd @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.bdd_below.preordering_bdd_axioms
thf(fact_7553_preordering__bdd_Oempty,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( condit622319405099724424ng_bdd @ A @ Less_eq @ Less )
=> ( condit16957441358409770ng_bdd @ A @ Less_eq @ ( bot_bot @ ( set @ A ) ) ) ) ).
% preordering_bdd.empty
thf(fact_7554_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_7555_numeral__sqr,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num] :
( ( numeral_numeral @ A @ ( sqr @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% numeral_sqr
thf(fact_7556_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R4: A > A > $o,A5: A,B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_7557_tranclp__trancl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_7558_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A8: A,B7: B] :
( ( R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( P @ A8 @ B7 ) )
=> ( ! [A8: A,B7: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( ( R3 @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A8 @ B7 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_7559_less__nat__rel,axiom,
( ( ord_less @ nat )
= ( transitive_tranclp @ nat
@ ^ [M2: nat,N2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% less_nat_rel
thf(fact_7560_tranclp__def,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R4: A > A > $o] :
( complete_lattice_lfp @ ( A > A > $o )
@ ^ [P6: A > A > $o,X12: A,X23: A] :
( ? [A5: A,B4: A] :
( ( X12 = A5 )
& ( X23 = B4 )
& ( R4 @ A5 @ B4 ) )
| ? [A5: A,B4: A,C5: A] :
( ( X12 = A5 )
& ( X23 = C5 )
& ( P6 @ A5 @ B4 )
& ( R4 @ B4 @ C5 ) ) ) ) ) ) ).
% tranclp_def
thf(fact_7561_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ).
% trancl_def
thf(fact_7562_reflp__refl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 ) ) ).
% reflp_refl_eq
thf(fact_7563_image2p__def,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType] :
( ( bNF_Greatest_image2p @ C @ A @ D @ B )
= ( ^ [F: C > A,G: D > B,R2: C > D > $o,X3: A,Y3: B] :
? [X9: C,Y8: D] :
( ( R2 @ X9 @ Y8 )
& ( ( F @ X9 )
= X3 )
& ( ( G @ Y8 )
= Y3 ) ) ) ) ).
% image2p_def
thf(fact_7564_reflp__sup,axiom,
! [A: $tType,R3: A > A > $o,S3: A > A > $o] :
( ( reflp @ A @ R3 )
=> ( ( reflp @ A @ S3 )
=> ( reflp @ A @ ( sup_sup @ ( A > A > $o ) @ R3 @ S3 ) ) ) ) ).
% reflp_sup
thf(fact_7565_Quotient__id__abs__transfer,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( reflp @ A @ R )
=> ( bNF_rel_fun @ A @ A @ A @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ T2
@ ^ [X3: A] : X3
@ Abs ) ) ) ).
% Quotient_id_abs_transfer
thf(fact_7566_reflp__inf,axiom,
! [A: $tType,R3: A > A > $o,S3: A > A > $o] :
( ( reflp @ A @ R3 )
=> ( ( reflp @ A @ S3 )
=> ( reflp @ A @ ( inf_inf @ ( A > A > $o ) @ R3 @ S3 ) ) ) ) ).
% reflp_inf
thf(fact_7567_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ code_pcr_integer )
@ ^ [M2: num,N2: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_7568_of__int__code_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( neg @ K ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_code(1)
thf(fact_7569_uminus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% uminus_integer.transfer
thf(fact_7570_integer_Oid__abs__transfer,axiom,
( bNF_rel_fun @ int @ int @ int @ code_integer
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ code_pcr_integer
@ ^ [X3: int] : X3
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_7571_integer_Orep__transfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [X3: int] : X3
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_7572_dup_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
@ code_dup ) ).
% dup.transfer
thf(fact_7573_one__integer_Otransfer,axiom,
code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% one_integer.transfer
thf(fact_7574_rtranclp__imp__Sup__relpowp,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A] :
( ( transitive_rtranclp @ A @ P @ X @ Y )
=> ( complete_Sup_Sup @ ( A > A > $o )
@ ( image2 @ nat @ ( A > A > $o )
@ ^ [N2: nat] : ( compow @ ( A > A > $o ) @ N2 @ P )
@ ( top_top @ ( set @ nat ) ) )
@ X
@ Y ) ) ).
% rtranclp_imp_Sup_relpowp
thf(fact_7575_rtranclp__is__Sup__relpowp,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [P2: A > A > $o] :
( complete_Sup_Sup @ ( A > A > $o )
@ ( image2 @ nat @ ( A > A > $o )
@ ^ [N2: nat] : ( compow @ ( A > A > $o ) @ N2 @ P2 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtranclp_is_Sup_relpowp
thf(fact_7576_rtranclp__reflclp,axiom,
! [A: $tType,R: A > A > $o] :
( ( transitive_rtranclp @ A
@ ( sup_sup @ ( A > A > $o ) @ R
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
= ( transitive_rtranclp @ A @ R ) ) ).
% rtranclp_reflclp
thf(fact_7577_rtranclp__reflclp__absorb,axiom,
! [A: $tType,R: A > A > $o] :
( ( sup_sup @ ( A > A > $o ) @ ( transitive_rtranclp @ A @ R )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( transitive_rtranclp @ A @ R ) ) ).
% rtranclp_reflclp_absorb
thf(fact_7578_reflclp__tranclp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( sup_sup @ ( A > A > $o ) @ ( transitive_tranclp @ A @ R3 )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( transitive_rtranclp @ A @ R3 ) ) ).
% reflclp_tranclp
thf(fact_7579_rtranclp__sup__rtranclp,axiom,
! [A: $tType,R: A > A > $o,S: A > A > $o] :
( ( transitive_rtranclp @ A @ ( sup_sup @ ( A > A > $o ) @ ( transitive_rtranclp @ A @ R ) @ ( transitive_rtranclp @ A @ S ) ) )
= ( transitive_rtranclp @ A @ ( sup_sup @ ( A > A > $o ) @ R @ S ) ) ) ).
% rtranclp_sup_rtranclp
thf(fact_7580_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R4: A > A > $o,X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_7581_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_7582_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_7583_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Bx @ By )
=> ( ! [A8: A,B7: B,Aa2: A,Ba: B] :
( ( R3 @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A8 @ B7 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_7584_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za2: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A8: A,B7: B] :
( ( R3 @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_7585_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A8: A,B7: B,Aa2: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A8 @ B7 ) )
=> ( ( R3 @ ( product_Pair @ A @ B @ A8 @ B7 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A8 @ B7 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_7586_rtranclp__r__diff__Id,axiom,
! [A: $tType,R3: A > A > $o] :
( ( transitive_rtranclp @ A
@ ( inf_inf @ ( A > A > $o ) @ R3
@ ^ [X3: A,Y3: A] : X3 != Y3 ) )
= ( transitive_rtranclp @ A @ R3 ) ) ).
% rtranclp_r_diff_Id
thf(fact_7587_rtranclp__def,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R4: A > A > $o] :
( complete_lattice_lfp @ ( A > A > $o )
@ ^ [P6: A > A > $o,X12: A,X23: A] :
( ? [A5: A] :
( ( X12 = A5 )
& ( X23 = A5 ) )
| ? [A5: A,B4: A,C5: A] :
( ( X12 = A5 )
& ( X23 = C5 )
& ( P6 @ A5 @ B4 )
& ( R4 @ B4 @ C5 ) ) ) ) ) ) ).
% rtranclp_def
thf(fact_7588_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% old.rec_bool_def
thf(fact_7589_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
thf(fact_7590_comm__monoid__list_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups1828464146339083142d_list @ A @ F2 @ Z2 )
=> ( comm_monoid @ A @ F2 @ Z2 ) ) ).
% comm_monoid_list.axioms(1)
thf(fact_7591_comm__monoid__list_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( ( groups_monoid_list @ A @ F2 @ Z2 )
=> ( groups1828464146339083142d_list @ A @ F2 @ Z2 ) ) ) ).
% comm_monoid_list.intro
thf(fact_7592_comm__monoid__list__def,axiom,
! [A: $tType] :
( ( groups1828464146339083142d_list @ A )
= ( ^ [F: A > A > A,Z5: A] :
( ( comm_monoid @ A @ F @ Z5 )
& ( groups_monoid_list @ A @ F @ Z5 ) ) ) ) ).
% comm_monoid_list_def
thf(fact_7593_monoid__list_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( groups_monoid_list @ A @ F2 @ Z2 ) ) ).
% monoid_list.intro
thf(fact_7594_monoid__list_Oaxioms,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups_monoid_list @ A @ F2 @ Z2 )
=> ( monoid @ A @ F2 @ Z2 ) ) ).
% monoid_list.axioms
thf(fact_7595_monoid__list__def,axiom,
! [A: $tType] :
( ( groups_monoid_list @ A )
= ( monoid @ A ) ) ).
% monoid_list_def
thf(fact_7596_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
thf(fact_7597_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: A > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A5: A,B4: B] : ( P @ A5 )
@ A4
@ B3 )
= ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ A4 ) )
=> ( P @ X3 ) ) ) ) ).
% laz_weak_Pa
thf(fact_7598_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: B > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A5: A] : P
@ A4
@ B3 )
= ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
& ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ B3 ) )
=> ( P @ X3 ) ) ) ) ).
% laz_weak_Pb
thf(fact_7599_laz__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q2: A > B > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
& ( Q2 @ X3 @ Y3 ) )
@ A3
@ B2 )
= ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
& ( list_all_zip @ A @ B @ Q2 @ A3 @ B2 ) ) ) ).
% laz_conj
thf(fact_7600_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As: list @ A,F2: C > B,Bs: list @ C] :
( ( list_all_zip @ A @ B @ P @ As @ ( map @ C @ B @ F2 @ Bs ) )
= ( list_all_zip @ A @ C
@ ^ [A5: A,B4: C] : ( P @ A5 @ ( F2 @ B4 ) )
@ As
@ Bs ) ) ).
% list_all_zip_map2
thf(fact_7601_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F2: C > A,As: list @ C,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F2 @ As ) @ Bs )
= ( list_all_zip @ C @ B
@ ^ [A5: C] : ( P @ ( F2 @ A5 ) )
@ As
@ Bs ) ) ).
% list_all_zip_map1
thf(fact_7602_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A5: A,B4: B] :
? [X4: C] : ( P @ A5 @ B4 @ X4 )
@ A4
@ B3 )
=> ~ ! [C8: list @ C] :
( ( list_all_zip @ A @ C
@ ^ [A5: A,C5: C] :
? [B4: B] : ( P @ A5 @ B4 @ C5 )
@ A4
@ C8 )
=> ~ ( list_all_zip @ B @ C
@ ^ [B4: B,C5: C] :
? [A5: A] : ( P @ A5 @ B4 @ C5 )
@ B3
@ C8 ) ) ) ).
% laz_swap_ex
thf(fact_7603_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A8: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A8 @ As4 ) )
=> ! [B7: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B7 @ Bs2 ) )
=> ( ( Y
= ( ( X @ A8 @ B7 )
& ( list_all_zip @ A @ B @ X @ As4 @ Bs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ ( cons @ B @ B7 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V3 @ Va ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
thf(fact_7604_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) )
=> ~ ! [A8: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A8 @ As4 ) )
=> ! [B7: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B7 @ Bs2 ) )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ ( cons @ B @ B7 @ Bs2 ) ) ) )
=> ~ ( ( X @ A8 @ B7 )
& ( list_all_zip @ A @ B @ X @ As4 @ Bs2 ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
thf(fact_7605_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ! [A8: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A8 @ As4 ) )
=> ! [B7: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B7 @ Bs2 ) )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A8 @ As4 ) @ ( cons @ B @ B7 @ Bs2 ) ) ) )
=> ( ( X @ A8 @ B7 )
& ( list_all_zip @ A @ B @ X @ As4 @ Bs2 ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V3 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
thf(fact_7606_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G2: C,Ga: B > C,F2: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G2 @ Ga ) @ ( map_option @ A @ B @ F2 ) )
= ( rec_option @ C @ A @ G2
@ ^ [X3: A] : ( Ga @ ( F2 @ X3 ) ) ) ) ).
% option.rec_o_map
thf(fact_7607_group_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( group_axioms @ A @ F2 @ Z2 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_7608_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F: A > A > A,Z5: A,Inverse2: A > A] :
( ! [A5: A] :
( ( F @ Z5 @ A5 )
= A5 )
& ! [A5: A] :
( ( F @ ( Inverse2 @ A5 ) @ A5 )
= Z5 ) ) ) ) ).
% group_axioms_def
thf(fact_7609_group__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ! [A8: A] :
( ( F2 @ Z2 @ A8 )
= A8 )
=> ( ! [A8: A] :
( ( F2 @ ( Inverse @ A8 ) @ A8 )
= Z2 )
=> ( group_axioms @ A @ F2 @ Z2 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_7610_group_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( semigroup @ A @ F2 )
=> ( ( group_axioms @ A @ F2 @ Z2 @ Inverse )
=> ( group @ A @ F2 @ Z2 @ Inverse ) ) ) ).
% group.intro
thf(fact_7611_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F: A > A > A,Z5: A,Inverse2: A > A] :
( ( semigroup @ A @ F )
& ( group_axioms @ A @ F @ Z5 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_7612_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_7613_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.semigroup_axioms
thf(fact_7614_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.semigroup_axioms
thf(fact_7615_semigroup_Oassoc,axiom,
! [A: $tType,F2: A > A > A,A3: A,B2: A,C2: A] :
( ( semigroup @ A @ F2 )
=> ( ( F2 @ ( F2 @ A3 @ B2 ) @ C2 )
= ( F2 @ A3 @ ( F2 @ B2 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_7616_semigroup_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A8: A,B7: A,C4: A] :
( ( F2 @ ( F2 @ A8 @ B7 ) @ C4 )
= ( F2 @ A8 @ ( F2 @ B7 @ C4 ) ) )
=> ( semigroup @ A @ F2 ) ) ).
% semigroup.intro
thf(fact_7617_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F: A > A > A] :
! [A5: A,B4: A,C5: A] :
( ( F @ ( F @ A5 @ B4 ) @ C5 )
= ( F @ A5 @ ( F @ B4 @ C5 ) ) ) ) ) ).
% semigroup_def
thf(fact_7618_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( semigroup @ A @ F2 ) ) ).
% monoid.axioms(1)
thf(fact_7619_group_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( semigroup @ A @ F2 ) ) ).
% group.axioms(1)
thf(fact_7620_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_7621_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_7622_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.semigroup_axioms
thf(fact_7623_monoid_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semigroup @ A @ F2 )
=> ( ( monoid_axioms @ A @ F2 @ Z2 )
=> ( monoid @ A @ F2 @ Z2 ) ) ) ).
% monoid.intro
thf(fact_7624_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F: A > A > A,Z5: A] :
( ( semigroup @ A @ F )
& ( monoid_axioms @ A @ F @ Z5 ) ) ) ) ).
% monoid_def
thf(fact_7625_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F: A > A > A,Z5: A] :
( ! [A5: A] :
( ( F @ Z5 @ A5 )
= A5 )
& ! [A5: A] :
( ( F @ A5 @ Z5 )
= A5 ) ) ) ) ).
% monoid_axioms_def
thf(fact_7626_monoid__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ! [A8: A] :
( ( F2 @ Z2 @ A8 )
= A8 )
=> ( ! [A8: A] :
( ( F2 @ A8 @ Z2 )
= A8 )
=> ( monoid_axioms @ A @ F2 @ Z2 ) ) ) ).
% monoid_axioms.intro
thf(fact_7627_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( monoid_axioms @ A @ F2 @ Z2 ) ) ).
% monoid.axioms(2)
thf(fact_7628_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% ordering_top.axioms(2)
thf(fact_7629_sum_Oinj__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F22: B > D] :
( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ B @ D @ F22 @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F22 ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ) ) ).
% sum.inj_map
thf(fact_7630_map__sum_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( sum_map_sum @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3 )
= ( id @ ( sum_sum @ A @ B ) ) ) ).
% map_sum.identity
thf(fact_7631_sum_Omap__ident,axiom,
! [B: $tType,A: $tType,T4: sum_sum @ A @ B] :
( ( sum_map_sum @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3
@ T4 )
= T4 ) ).
% sum.map_ident
thf(fact_7632_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Top: A] :
( ! [A8: A] : ( Less_eq @ A8 @ Top )
=> ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% ordering_top_axioms.intro
thf(fact_7633_ordering__top__axioms__def,axiom,
! [A: $tType] :
( ( ordering_top_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Top2: A] :
! [A5: A] : ( Less_eq2 @ A5 @ Top2 ) ) ) ).
% ordering_top_axioms_def
thf(fact_7634_sum_Orel__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A18: set @ A,F1: A > C,A25: set @ B,F22: B > D] :
( ( bNF_rel_sum @ A @ C @ B @ D @ ( bNF_Grp @ A @ C @ A18 @ F1 ) @ ( bNF_Grp @ B @ D @ A25 @ F22 ) )
= ( bNF_Grp @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D )
@ ( collect @ ( sum_sum @ A @ B )
@ ^ [X3: sum_sum @ A @ B] :
( ( ord_less_eq @ ( set @ A ) @ ( basic_setl @ A @ B @ X3 ) @ A18 )
& ( ord_less_eq @ ( set @ B ) @ ( basic_setr @ A @ B @ X3 ) @ A25 ) ) )
@ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F22 ) ) ) ).
% sum.rel_Grp
thf(fact_7635_sum_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R15: A > C > $o,R25: B > D > $o,A5: sum_sum @ A @ B,B4: sum_sum @ C @ D] :
? [Z5: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z5
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X3: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R15 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z5 )
= A5 )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z5 )
= B4 ) ) ) ) ).
% sum.in_rel
thf(fact_7636_sum_Orel__map_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,S1b: E > C > $o,S2b: F4 > D > $o,I1: A > E,I22: B > F4,X: sum_sum @ A @ B,Y: sum_sum @ C @ D] :
( ( bNF_rel_sum @ E @ C @ F4 @ D @ S1b @ S2b @ ( sum_map_sum @ A @ E @ B @ F4 @ I1 @ I22 @ X ) @ Y )
= ( bNF_rel_sum @ A @ C @ B @ D
@ ^ [X3: A] : ( S1b @ ( I1 @ X3 ) )
@ ^ [X3: B] : ( S2b @ ( I22 @ X3 ) )
@ X
@ Y ) ) ).
% sum.rel_map(1)
thf(fact_7637_sum_Orel__map_I2_J,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,S1a: A > E > $o,S2a: B > F4 > $o,X: sum_sum @ A @ B,G1: C > E,G22: D > F4,Y: sum_sum @ C @ D] :
( ( bNF_rel_sum @ A @ E @ B @ F4 @ S1a @ S2a @ X @ ( sum_map_sum @ C @ E @ D @ F4 @ G1 @ G22 @ Y ) )
= ( bNF_rel_sum @ A @ C @ B @ D
@ ^ [X3: A,Y3: C] : ( S1a @ X3 @ ( G1 @ Y3 ) )
@ ^ [X3: B,Y3: D] : ( S2a @ X3 @ ( G22 @ Y3 ) )
@ X
@ Y ) ) ).
% sum.rel_map(2)
thf(fact_7638_sum_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R15: A > C > $o,R25: B > D > $o] :
( relcompp @ ( sum_sum @ A @ B ) @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( conversep @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X3: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R15 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X3: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R15 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X3 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% sum.rel_compp_Grp
thf(fact_7639_ordering__top__def,axiom,
! [A: $tType] :
( ( ordering_top @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,Top2: A] :
( ( ordering @ A @ Less_eq2 @ Less2 )
& ( ordering_top_axioms @ A @ Less_eq2 @ Top2 ) ) ) ) ).
% ordering_top_def
thf(fact_7640_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > 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
thf(fact_7641_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A3 = B2 )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ B2 @ A3 ) ) ) ) ).
% ordering.eq_iff
thf(fact_7642_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
thf(fact_7643_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( Less @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_7644_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_7645_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_7646_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A3 != B2 )
=> ( ( Less_eq @ A3 @ B2 )
=> ( Less @ A3 @ B2 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_7647_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A8: A,B7: A] :
( ( Less_eq @ A8 @ B7 )
= ( ( Less @ A8 @ B7 )
| ( A8 = B7 ) ) )
=> ( ! [A8: A,B7: A] :
( ( Less @ A8 @ B7 )
=> ~ ( Less @ B7 @ A8 ) )
=> ( ! [A8: A] :
~ ( Less @ A8 @ A8 )
=> ( ! [A8: A,B7: A,C4: A] :
( ( Less @ A8 @ B7 )
=> ( ( Less @ B7 @ C4 )
=> ( Less @ A8 @ C4 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_7648_ordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A5: A,B4: A] : ( Less_eq @ B4 @ A5 )
@ ^ [A5: A,B4: A] : ( Less @ B4 @ A5 ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_dualI
thf(fact_7649_gcd__nat_Oordering__axioms,axiom,
( ordering @ nat @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.ordering_axioms
thf(fact_7650_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_top.axioms(1)
thf(fact_7651_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_7652_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_7653_subset__mset_Odual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ordering @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.dual_order.ordering_axioms
thf(fact_7654_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F2: A > B,G2: B > A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X2: A] :
( ( G2 @ ( F2 @ X2 ) )
= X2 )
=> ( order_strict_mono @ B @ A @ G2 ) ) ) ) ) ).
% strict_mono_inv
thf(fact_7655_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F2: code_natural > A,N: code_natural,M: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ N @ M ) ) )
=> ( ( iterate_upto @ A @ F2 @ N @ M )
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M @ N ) @ ( empty @ A ) @ ( insert @ A @ ( F2 @ N ) @ ( iterate_upto @ A @ F2 @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M ) ) ) ) ) ) ).
% Predicate.iterate_upto.psimps
thf(fact_7656_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_7657_Predicate_Oiterate__upto_Oelims,axiom,
! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
( ( ( iterate_upto @ A @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.elims
thf(fact_7658_Predicate_Oiterate__upto_Osimps,axiom,
! [A: $tType] :
( ( iterate_upto @ A )
= ( ^ [F: code_natural > A,N2: code_natural,M2: code_natural] :
( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M2 @ N2 ) @ ( empty @ A ) @ ( insert @ A @ ( F @ N2 ) @ ( iterate_upto @ A @ F @ ( plus_plus @ code_natural @ N2 @ ( one_one @ code_natural ) ) @ M2 ) ) ) ) ) ) ).
% Predicate.iterate_upto.simps
thf(fact_7659_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_7660_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( order_strict_mono @ A @ B @ F2 ) ) ) ).
% strict_monoI
thf(fact_7661_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F: A > B] :
! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_7662_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_7663_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% strict_mono_mono
thf(fact_7664_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_7665_bot__set__code,axiom,
! [A: $tType] :
( ( bot_bot @ ( pred @ A ) )
= ( seq2 @ A
@ ^ [U2: product_unit] : ( empty @ A ) ) ) ).
% bot_set_code
thf(fact_7666_Predicate_Osingle__code,axiom,
! [A: $tType] :
( ( single @ A )
= ( ^ [X3: A] :
( seq2 @ A
@ ^ [U2: product_unit] : ( insert @ A @ X3 @ ( bot_bot @ ( pred @ A ) ) ) ) ) ) ).
% Predicate.single_code
thf(fact_7667_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
( ( ( iterate_upto @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.pelims
thf(fact_7668_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType] :
( ( random_iterate_upto @ A )
= ( ^ [F: code_natural > A,N2: code_natural,M2: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F @ N2 @ M2 ) ) ) ) ).
% Random_Pred.iterate_upto_def
thf(fact_7669_bind__code,axiom,
! [B: $tType,A: $tType,G2: product_unit > ( seq @ B ),F2: B > ( pred @ A )] :
( ( bind2 @ B @ A @ ( seq2 @ B @ G2 ) @ F2 )
= ( seq2 @ A
@ ^ [U2: product_unit] : ( apply @ B @ A @ F2 @ ( G2 @ product_Unity ) ) ) ) ).
% bind_code
thf(fact_7670_strict__mono__inv__on__range,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( strict_mono_on @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% strict_mono_inv_on_range
thf(fact_7671_pred__of__seq_Osimps_I2_J,axiom,
! [A: $tType,X: A,P: pred @ A] :
( ( pred_of_seq @ A @ ( insert @ A @ X @ P ) )
= ( sup_sup @ ( pred @ A ) @ ( single @ A @ X ) @ P ) ) ).
% pred_of_seq.simps(2)
thf(fact_7672_pred__of__seq_Osimps_I1_J,axiom,
! [A: $tType] :
( ( pred_of_seq @ A @ ( empty @ A ) )
= ( bot_bot @ ( pred @ A ) ) ) ).
% pred_of_seq.simps(1)
thf(fact_7673_adjunct__sup,axiom,
! [A: $tType,P: pred @ A,Xq: seq @ A] :
( ( pred_of_seq @ A @ ( adjunct @ A @ P @ Xq ) )
= ( sup_sup @ ( pred @ A ) @ P @ ( pred_of_seq @ A @ Xq ) ) ) ).
% adjunct_sup
thf(fact_7674_adjunct_Osimps_I2_J,axiom,
! [A: $tType,P: pred @ A,X: A,Q2: pred @ A] :
( ( adjunct @ A @ P @ ( insert @ A @ X @ Q2 ) )
= ( insert @ A @ X @ ( sup_sup @ ( pred @ A ) @ Q2 @ P ) ) ) ).
% adjunct.simps(2)
thf(fact_7675_sup__code,axiom,
! [A: $tType,F2: product_unit > ( seq @ A ),G2: product_unit > ( seq @ A )] :
( ( sup_sup @ ( pred @ A ) @ ( seq2 @ A @ F2 ) @ ( seq2 @ A @ G2 ) )
= ( seq2 @ A
@ ^ [U2: product_unit] :
( case_seq @ ( seq @ A ) @ A @ ( G2 @ product_Unity )
@ ^ [X3: A,P2: pred @ A] : ( insert @ A @ X3 @ ( sup_sup @ ( pred @ A ) @ P2 @ ( seq2 @ A @ G2 ) ) )
@ ^ [P2: pred @ A,Xq2: seq @ A] : ( adjunct @ A @ ( seq2 @ A @ G2 ) @ ( join @ A @ P2 @ Xq2 ) )
@ ( F2 @ product_Unity ) ) ) ) ).
% sup_code
thf(fact_7676_of__seq__code_I1_J,axiom,
! [A: $tType] :
( ( set_of_seq @ A @ ( empty @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% of_seq_code(1)
thf(fact_7677_seq_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H3: B > C,F1: B,F22: A > ( pred @ A ) > B,F32: ( pred @ A ) > ( seq @ A ) > B,Seq: seq @ A] :
( ( H3 @ ( case_seq @ B @ A @ F1 @ F22 @ F32 @ Seq ) )
= ( case_seq @ C @ A @ ( H3 @ F1 )
@ ^ [X12: A,X23: pred @ A] : ( H3 @ ( F22 @ X12 @ X23 ) )
@ ^ [X12: pred @ A,X23: seq @ A] : ( H3 @ ( F32 @ X12 @ X23 ) )
@ Seq ) ) ).
% seq.case_distrib
thf(fact_7678_pred__of__seq_Osimps_I3_J,axiom,
! [A: $tType,P: pred @ A,Xq: seq @ A] :
( ( pred_of_seq @ A @ ( join @ A @ P @ Xq ) )
= ( sup_sup @ ( pred @ A ) @ P @ ( pred_of_seq @ A @ Xq ) ) ) ).
% pred_of_seq.simps(3)
thf(fact_7679_of__seq__code_I3_J,axiom,
! [B: $tType,P: pred @ B,Xq: seq @ B] :
( ( set_of_seq @ B @ ( join @ B @ P @ Xq ) )
= ( sup_sup @ ( set @ B ) @ ( set_of_pred @ B @ P ) @ ( set_of_seq @ B @ Xq ) ) ) ).
% of_seq_code(3)
thf(fact_7680_of__pred__code,axiom,
! [A: $tType,F2: product_unit > ( seq @ A )] :
( ( set_of_pred @ A @ ( seq2 @ A @ F2 ) )
= ( case_seq @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X3: A,P2: pred @ A] : ( insert2 @ A @ X3 @ ( set_of_pred @ A @ P2 ) )
@ ^ [P2: pred @ A,Xq2: seq @ A] : ( sup_sup @ ( set @ A ) @ ( set_of_pred @ A @ P2 ) @ ( set_of_seq @ A @ Xq2 ) )
@ ( F2 @ product_Unity ) ) ) ).
% of_pred_code
thf(fact_7681_less__eq__pred__code,axiom,
! [A: $tType,F2: product_unit > ( seq @ A ),Q2: pred @ A] :
( ( ord_less_eq @ ( pred @ A ) @ ( seq2 @ A @ F2 ) @ Q2 )
= ( case_seq @ $o @ A @ $true
@ ^ [X3: A,P2: pred @ A] :
( ( eval @ A @ Q2 @ X3 )
& ( ord_less_eq @ ( pred @ A ) @ P2 @ Q2 ) )
@ ^ [P2: pred @ A,Xq2: seq @ A] :
( ( ord_less_eq @ ( pred @ A ) @ P2 @ Q2 )
& ( contained @ A @ Xq2 @ Q2 ) )
@ ( F2 @ product_Unity ) ) ) ).
% less_eq_pred_code
thf(fact_7682_rec__natural__def,axiom,
! [T: $tType] :
( ( code_rec_natural @ T )
= ( ^ [F12: T,F23: code_natural > T > T,X3: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% rec_natural_def
thf(fact_7683_case__natural__def,axiom,
! [T: $tType] :
( ( code_case_natural @ T )
= ( ^ [F12: T,F23: code_natural > T] :
( code_rec_natural @ T @ F12
@ ^ [X12: code_natural,X23: T] : ( F23 @ X12 ) ) ) ) ).
% case_natural_def
thf(fact_7684_the__only_Osimps_I2_J,axiom,
! [A: $tType,P: pred @ A,Default: product_unit > A,X: A] :
( ( ( is_empty @ A @ P )
=> ( ( the_only @ A @ Default @ ( insert @ A @ X @ P ) )
= X ) )
& ( ~ ( is_empty @ A @ P )
=> ( ( ( X
= ( singleton @ A @ Default @ P ) )
=> ( ( the_only @ A @ Default @ ( insert @ A @ X @ P ) )
= X ) )
& ( ( X
!= ( singleton @ A @ Default @ P ) )
=> ( ( the_only @ A @ Default @ ( insert @ A @ X @ P ) )
= ( Default @ product_Unity ) ) ) ) ) ) ).
% the_only.simps(2)
thf(fact_7685_Predicate_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: pred @ A] :
( A6
= ( bot_bot @ ( pred @ A ) ) ) ) ) ).
% Predicate.is_empty_def
thf(fact_7686_is__empty__bot,axiom,
! [A: $tType] : ( is_empty @ A @ ( bot_bot @ ( pred @ A ) ) ) ).
% is_empty_bot
thf(fact_7687_is__empty__sup,axiom,
! [A: $tType,A4: pred @ A,B3: pred @ A] :
( ( is_empty @ A @ ( sup_sup @ ( pred @ A ) @ A4 @ B3 ) )
= ( ( is_empty @ A @ A4 )
& ( is_empty @ A @ B3 ) ) ) ).
% is_empty_sup
thf(fact_7688_singleton__code,axiom,
! [A: $tType,Default: product_unit > A,F2: product_unit > ( seq @ A )] :
( ( singleton @ A @ Default @ ( seq2 @ A @ F2 ) )
= ( case_seq @ A @ A @ ( Default @ product_Unity )
@ ^ [X3: A,P2: pred @ A] :
( if @ A @ ( is_empty @ A @ P2 ) @ X3
@ ( if @ A
@ ( X3
= ( singleton @ A @ Default @ P2 ) )
@ X3
@ ( Default @ product_Unity ) ) )
@ ^ [P2: pred @ A,Xq2: seq @ A] :
( if @ A @ ( is_empty @ A @ P2 ) @ ( the_only @ A @ Default @ Xq2 )
@ ( if @ A @ ( null @ A @ Xq2 ) @ ( singleton @ A @ Default @ P2 )
@ ( if @ A
@ ( ( singleton @ A @ Default @ P2 )
= ( the_only @ A @ Default @ Xq2 ) )
@ ( singleton @ A @ Default @ P2 )
@ ( Default @ product_Unity ) ) ) )
@ ( F2 @ product_Unity ) ) ) ).
% singleton_code
thf(fact_7689_the__only_Osimps_I3_J,axiom,
! [A: $tType,P: pred @ A,Default: product_unit > A,Xq: seq @ A] :
( ( ( is_empty @ A @ P )
=> ( ( the_only @ A @ Default @ ( join @ A @ P @ Xq ) )
= ( the_only @ A @ Default @ Xq ) ) )
& ( ~ ( is_empty @ A @ P )
=> ( ( ( null @ A @ Xq )
=> ( ( the_only @ A @ Default @ ( join @ A @ P @ Xq ) )
= ( singleton @ A @ Default @ P ) ) )
& ( ~ ( null @ A @ Xq )
=> ( ( the_only @ A @ Default @ ( join @ A @ P @ Xq ) )
= ( if @ A
@ ( ( singleton @ A @ Default @ P )
= ( the_only @ A @ Default @ Xq ) )
@ ( singleton @ A @ Default @ P )
@ ( Default @ product_Unity ) ) ) ) ) ) ) ).
% the_only.simps(3)
thf(fact_7690_Abs__int__cases,axiom,
! [X: int] :
~ ! [Y2: set @ ( product_prod @ nat @ nat )] :
( ( X
= ( abs_int @ Y2 ) )
=> ~ ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y2
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) ) ) ).
% Abs_int_cases
thf(fact_7691_Abs__int__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y2: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y2
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) )
=> ( P @ ( abs_int @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_int_induct
thf(fact_7692_Abs__int__inject,axiom,
! [X: set @ ( product_prod @ nat @ nat ),Y: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ X
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) )
=> ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) )
=> ( ( ( abs_int @ X )
= ( abs_int @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_int_inject
thf(fact_7693_Abs__int__inverse,axiom,
! [Y: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) )
=> ( ( rep_int @ ( abs_int @ Y ) )
= Y ) ) ).
% Abs_int_inverse
thf(fact_7694_type__definition__int,axiom,
( type_definition @ int @ ( set @ ( product_prod @ nat @ nat ) ) @ rep_int @ abs_int
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) ) ).
% type_definition_int
thf(fact_7695_Rep__int__induct,axiom,
! [Y: set @ ( product_prod @ nat @ nat ),P: ( set @ ( product_prod @ nat @ nat ) ) > $o] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) )
=> ( ! [X2: int] : ( P @ ( rep_int @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_int_induct
thf(fact_7696_Rep__int__cases,axiom,
! [Y: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) )
=> ~ ! [X2: int] :
( Y
!= ( rep_int @ X2 ) ) ) ).
% Rep_int_cases
thf(fact_7697_Rep__int,axiom,
! [X: int] :
( member @ ( set @ ( product_prod @ nat @ nat ) ) @ ( rep_int @ X )
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C5: set @ ( product_prod @ nat @ nat )] :
? [X3: product_prod @ nat @ nat] :
( ( intrel @ X3 @ X3 )
& ( C5
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X3 ) ) ) ) ) ) ).
% Rep_int
thf(fact_7698_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ).
% ordering.axioms(2)
thf(fact_7699_Int_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(9)
thf(fact_7700_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A8: A,B7: A] :
( ( Less @ A8 @ B7 )
= ( ( Less_eq @ A8 @ B7 )
& ( A8 != B7 ) ) )
=> ( ! [A8: A,B7: A] :
( ( Less_eq @ A8 @ B7 )
=> ( ( Less_eq @ B7 @ A8 )
=> ( A8 = B7 ) ) )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% ordering_axioms.intro
thf(fact_7701_ordering__axioms__def,axiom,
! [A: $tType] :
( ( ordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less2 @ A5 @ B4 )
= ( ( Less_eq2 @ A5 @ B4 )
& ( A5 != B4 ) ) )
& ! [A5: A,B4: A] :
( ( Less_eq2 @ A5 @ B4 )
=> ( ( Less_eq2 @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ) ) ) ).
% ordering_axioms_def
thf(fact_7702_Int_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(8)
thf(fact_7703_ordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( ordering_axioms @ A @ Less_eq @ Less )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ).
% ordering.intro
thf(fact_7704_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( partial_preordering @ A @ Less_eq2 )
& ( ordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% ordering_def
thf(fact_7705_subset__mset_Odual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( partial_preordering @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.dual_order.partial_preordering_axioms
thf(fact_7706_partial__preordering__def,axiom,
! [A: $tType] :
( ( partial_preordering @ A )
= ( ^ [Less_eq2: A > A > $o] :
( ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
& ! [A5: A,B4: A,C5: A] :
( ( Less_eq2 @ A5 @ B4 )
=> ( ( Less_eq2 @ B4 @ C5 )
=> ( Less_eq2 @ A5 @ C5 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_7707_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: A,B2: A,C2: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ A3 @ C2 ) ) ) ) ).
% partial_preordering.trans
thf(fact_7708_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o] :
( ! [A8: A] : ( Less_eq @ A8 @ A8 )
=> ( ! [A8: A,B7: A,C4: A] :
( ( Less_eq @ A8 @ B7 )
=> ( ( Less_eq @ B7 @ C4 )
=> ( Less_eq @ A8 @ C4 ) ) )
=> ( partial_preordering @ A @ Less_eq ) ) ) ).
% partial_preordering.intro
thf(fact_7709_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( Less_eq @ A3 @ A3 ) ) ).
% partial_preordering.refl
thf(fact_7710_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A @ Less_eq @ Less )
=> ( partial_preordering @ A @ Less_eq ) ) ).
% ordering.axioms(1)
thf(fact_7711_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_7712_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% order.partial_preordering_axioms
thf(fact_7713_semilattice__neutr__def,axiom,
! [A: $tType] :
( ( semilattice_neutr @ A )
= ( ^ [F: A > A > A,Z5: A] :
( ( semilattice @ A @ F )
& ( comm_monoid @ A @ F @ Z5 ) ) ) ) ).
% semilattice_neutr_def
thf(fact_7714_semilattice__neutr_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semilattice @ A @ F2 )
=> ( ( comm_monoid @ A @ F2 @ Z2 )
=> ( semilattice_neutr @ A @ F2 @ Z2 ) ) ) ).
% semilattice_neutr.intro
thf(fact_7715_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( semilattice @ A @ F2 ) ) ).
% semilattice_order.axioms(1)
thf(fact_7716_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice @ A @ ( sup_sup @ A ) ) ) ).
% sup.semilattice_axioms
thf(fact_7717_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_min @ A ) ) ) ).
% min.semilattice_axioms
thf(fact_7718_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_max @ A ) ) ) ).
% max.semilattice_axioms
thf(fact_7719_semilattice_Oidem,axiom,
! [A: $tType,F2: A > A > A,A3: A] :
( ( semilattice @ A @ F2 )
=> ( ( F2 @ A3 @ A3 )
= A3 ) ) ).
% semilattice.idem
thf(fact_7720_semilattice_Oleft__idem,axiom,
! [A: $tType,F2: A > A > A,A3: A,B2: A] :
( ( semilattice @ A @ F2 )
=> ( ( F2 @ A3 @ ( F2 @ A3 @ B2 ) )
= ( F2 @ A3 @ B2 ) ) ) ).
% semilattice.left_idem
thf(fact_7721_semilattice_Oright__idem,axiom,
! [A: $tType,F2: A > A > A,A3: A,B2: A] :
( ( semilattice @ A @ F2 )
=> ( ( F2 @ ( F2 @ A3 @ B2 ) @ B2 )
= ( F2 @ A3 @ B2 ) ) ) ).
% semilattice.right_idem
thf(fact_7722_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semilattice_neutr @ A @ F2 @ Z2 )
=> ( semilattice @ A @ F2 ) ) ).
% semilattice_neutr.axioms(1)
thf(fact_7723_semilattice__map2,axiom,
! [A: $tType,F2: A > A > A] :
( ( semilattice @ A @ F2 )
=> ( semilattice @ ( list @ A )
@ ^ [Xs2: list @ A,Ys2: list @ A] : ( map @ ( product_prod @ A @ A ) @ A @ ( product_case_prod @ A @ A @ A @ F2 ) @ ( zip @ A @ A @ Xs2 @ Ys2 ) ) ) ) ).
% semilattice_map2
thf(fact_7724_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice @ A @ ( inf_inf @ A ) ) ) ).
% inf.semilattice_axioms
thf(fact_7725_semilattice__order__def,axiom,
! [A: $tType] :
( ( semilattice_order @ A )
= ( ^ [F: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice @ A @ F )
& ( semila6385135966242565138axioms @ A @ F @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_order_def
thf(fact_7726_semilattice__order_Ointro,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice @ A @ F2 )
=> ( ( semila6385135966242565138axioms @ A @ F2 @ Less_eq @ Less )
=> ( semilattice_order @ A @ F2 @ Less_eq @ Less ) ) ) ).
% semilattice_order.intro
thf(fact_7727_semilattice__order__axioms__def,axiom,
! [A: $tType] :
( ( semila6385135966242565138axioms @ A )
= ( ^ [F: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less_eq2 @ A5 @ B4 )
= ( A5
= ( F @ A5 @ B4 ) ) )
& ! [A5: A,B4: A] :
( ( Less2 @ A5 @ B4 )
= ( ( A5
= ( F @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ) ).
% semilattice_order_axioms_def
thf(fact_7728_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,F2: A > A > A,Less: A > A > $o] :
( ! [A8: A,B7: A] :
( ( Less_eq @ A8 @ B7 )
= ( A8
= ( F2 @ A8 @ B7 ) ) )
=> ( ! [A8: A,B7: A] :
( ( Less @ A8 @ B7 )
= ( ( A8
= ( F2 @ A8 @ B7 ) )
& ( A8 != B7 ) ) )
=> ( semila6385135966242565138axioms @ A @ F2 @ Less_eq @ Less ) ) ) ).
% semilattice_order_axioms.intro
thf(fact_7729_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( semila6385135966242565138axioms @ A @ F2 @ Less_eq @ Less ) ) ).
% semilattice_order.axioms(2)
thf(fact_7730_subset__mset_Odual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( preordering @ ( multiset @ A )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subseteq_mset @ A @ Y3 @ X3 )
@ ^ [X3: multiset @ A,Y3: multiset @ A] : ( subset_mset @ A @ Y3 @ X3 ) ) ).
% subset_mset.dual_order.preordering_axioms
thf(fact_7731_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( comm_monoid_axioms @ A @ F2 @ Z2 ) ) ).
% comm_monoid.axioms(2)
thf(fact_7732_order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.preordering_axioms
thf(fact_7733_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ! [A8: A] :
( ( F2 @ A8 @ Z2 )
= A8 )
=> ( comm_monoid_axioms @ A @ F2 @ Z2 ) ) ).
% comm_monoid_axioms.intro
thf(fact_7734_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F: A > A > A,Z5: A] :
! [A5: A] :
( ( F @ A5 @ Z5 )
= A5 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_7735_preordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ~ ( Less @ B2 @ A3 ) ) ) ).
% preordering.asym
thf(fact_7736_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ~ ( Less @ A3 @ A3 ) ) ).
% preordering.irrefl
thf(fact_7737_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ A3 @ C2 ) ) ) ) ).
% preordering.strict_trans
thf(fact_7738_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ A3 @ C2 ) ) ) ) ).
% preordering.strict_trans1
thf(fact_7739_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less @ A3 @ C2 ) ) ) ) ).
% preordering.strict_trans2
thf(fact_7740_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq @ A3 @ B2 )
& ~ ( Less_eq @ B2 @ A3 ) ) ) ) ).
% preordering.strict_iff_not
thf(fact_7741_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% preordering.strict_implies_order
thf(fact_7742_preordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A8: A,B7: A] :
( ( Less_eq @ A8 @ B7 )
= ( ( Less @ A8 @ B7 )
| ( A8 = B7 ) ) )
=> ( ! [A8: A,B7: A] :
( ( Less @ A8 @ B7 )
=> ~ ( Less @ B7 @ A8 ) )
=> ( ! [A8: A] :
~ ( Less @ A8 @ A8 )
=> ( ! [A8: A,B7: A,C4: A] :
( ( Less @ A8 @ B7 )
=> ( ( Less @ B7 @ C4 )
=> ( Less @ A8 @ C4 ) ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_7743_preordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A
@ ^ [A5: A,B4: A] : ( Less_eq @ B4 @ A5 )
@ ^ [A5: A,B4: A] : ( Less @ B4 @ A5 ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ).
% preordering_dualI
thf(fact_7744_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A @ Less_eq @ Less )
=> ( partial_preordering @ A @ Less_eq ) ) ).
% preordering.axioms(1)
thf(fact_7745_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_7746_preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( preordering_axioms @ A @ Less_eq @ Less )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ).
% preordering.intro
thf(fact_7747_preordering__def,axiom,
! [A: $tType] :
( ( preordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( partial_preordering @ A @ Less_eq2 )
& ( preordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% preordering_def
thf(fact_7748_gcd__nat_Opreordering__axioms,axiom,
( preordering @ nat @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.preordering_axioms
thf(fact_7749_preordering__axioms__def,axiom,
! [A: $tType] :
( ( preordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
! [A5: A,B4: A] :
( ( Less2 @ A5 @ B4 )
= ( ( Less_eq2 @ A5 @ B4 )
& ~ ( Less_eq2 @ B4 @ A5 ) ) ) ) ) ).
% preordering_axioms_def
thf(fact_7750_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A8: A,B7: A] :
( ( Less @ A8 @ B7 )
= ( ( Less_eq @ A8 @ B7 )
& ~ ( Less_eq @ B7 @ A8 ) ) )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering_axioms.intro
thf(fact_7751_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A @ Less_eq @ Less )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering.axioms(2)
thf(fact_7752_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F: A > A > A,Z5: A] :
( ( abel_semigroup @ A @ F )
& ( comm_monoid_axioms @ A @ F @ Z5 ) ) ) ) ).
% comm_monoid_def
thf(fact_7753_comm__monoid_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( comm_monoid_axioms @ A @ F2 @ Z2 )
=> ( comm_monoid @ A @ F2 @ Z2 ) ) ) ).
% comm_monoid.intro
thf(fact_7754_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A] :
( ( semilattice @ A @ F2 )
=> ( abel_semigroup @ A @ F2 ) ) ).
% semilattice.axioms(1)
thf(fact_7755_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( abel_semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.abel_semigroup_axioms
thf(fact_7756_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( abel_semigroup @ A @ Conj ) ) ).
% abstract_boolean_algebra.axioms(1)
thf(fact_7757_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( abel_semigroup @ A @ Disj ) ) ).
% abstract_boolean_algebra.axioms(2)
thf(fact_7758_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.abel_semigroup_axioms
thf(fact_7759_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.abel_semigroup_axioms
thf(fact_7760_abel__semigroup_Ocommute,axiom,
! [A: $tType,F2: A > A > A,A3: A,B2: A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( F2 @ A3 @ B2 )
= ( F2 @ B2 @ A3 ) ) ) ).
% abel_semigroup.commute
thf(fact_7761_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F2: A > A > A,B2: A,A3: A,C2: A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( F2 @ B2 @ ( F2 @ A3 @ C2 ) )
= ( F2 @ A3 @ ( F2 @ B2 @ C2 ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_7762_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( abel_semigroup @ A @ F2 ) ) ).
% comm_monoid.axioms(1)
thf(fact_7763_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_7764_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_7765_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( abel_semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.abel_semigroup_axioms
thf(fact_7766_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A] :
( ( abel_semigroup @ A @ F2 )
=> ( semigroup @ A @ F2 ) ) ).
% abel_semigroup.axioms(1)
thf(fact_7767_abel__semigroup__def,axiom,
! [A: $tType] :
( ( abel_semigroup @ A )
= ( ^ [F: A > A > A] :
( ( semigroup @ A @ F )
& ( abel_s757365448890700780axioms @ A @ F ) ) ) ) ).
% abel_semigroup_def
thf(fact_7768_abel__semigroup_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ( semigroup @ A @ F2 )
=> ( ( abel_s757365448890700780axioms @ A @ F2 )
=> ( abel_semigroup @ A @ F2 ) ) ) ).
% abel_semigroup.intro
thf(fact_7769_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A] :
( ( abel_semigroup @ A @ F2 )
=> ( abel_s757365448890700780axioms @ A @ F2 ) ) ).
% abel_semigroup.axioms(2)
thf(fact_7770_abel__semigroup__axioms__def,axiom,
! [A: $tType] :
( ( abel_s757365448890700780axioms @ A )
= ( ^ [F: A > A > A] :
! [A5: A,B4: A] :
( ( F @ A5 @ B4 )
= ( F @ B4 @ A5 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_7771_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A8: A,B7: A] :
( ( F2 @ A8 @ B7 )
= ( F2 @ B7 @ A8 ) )
=> ( abel_s757365448890700780axioms @ A @ F2 ) ) ).
% abel_semigroup_axioms.intro
thf(fact_7772_semilattice__def,axiom,
! [A: $tType] :
( ( semilattice @ A )
= ( ^ [F: A > A > A] :
( ( abel_semigroup @ A @ F )
& ( semilattice_axioms @ A @ F ) ) ) ) ).
% semilattice_def
thf(fact_7773_semilattice_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( semilattice_axioms @ A @ F2 )
=> ( semilattice @ A @ F2 ) ) ) ).
% semilattice.intro
thf(fact_7774_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A] :
( ( semilattice @ A @ F2 )
=> ( semilattice_axioms @ A @ F2 ) ) ).
% semilattice.axioms(2)
thf(fact_7775_semilattice__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A8: A] :
( ( F2 @ A8 @ A8 )
= A8 )
=> ( semilattice_axioms @ A @ F2 ) ) ).
% semilattice_axioms.intro
thf(fact_7776_semilattice__axioms__def,axiom,
! [A: $tType] :
( ( semilattice_axioms @ A )
= ( ^ [F: A > A > A] :
! [A5: A] :
( ( F @ A5 @ A5 )
= A5 ) ) ) ).
% semilattice_axioms_def
thf(fact_7777_abstract__boolean__algebra__def,axiom,
! [A: $tType] :
( ( boolea2506097494486148201lgebra @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One2: A] :
( ( abel_semigroup @ A @ Conj2 )
& ( abel_semigroup @ A @ Disj2 )
& ( boolea6902313364301356556axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One2 ) ) ) ) ).
% abstract_boolean_algebra_def
thf(fact_7778_abstract__boolean__algebra_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: 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
thf(fact_7779_abstract__boolean__algebra__axioms__def,axiom,
! [A: $tType] :
( ( boolea6902313364301356556axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One2: A] :
( ! [X3: A,Y3: A,Z5: A] :
( ( Conj2 @ X3 @ ( Disj2 @ Y3 @ Z5 ) )
= ( Disj2 @ ( Conj2 @ X3 @ Y3 ) @ ( Conj2 @ X3 @ Z5 ) ) )
& ! [X3: A,Y3: A,Z5: A] :
( ( Disj2 @ X3 @ ( Conj2 @ Y3 @ Z5 ) )
= ( Conj2 @ ( Disj2 @ X3 @ Y3 ) @ ( Disj2 @ X3 @ Z5 ) ) )
& ! [X3: A] :
( ( Conj2 @ X3 @ One2 )
= X3 )
& ! [X3: A] :
( ( Disj2 @ X3 @ Zero2 )
= X3 )
& ! [X3: A] :
( ( Conj2 @ X3 @ ( Compl2 @ X3 ) )
= Zero2 )
& ! [X3: A] :
( ( Disj2 @ X3 @ ( Compl2 @ X3 ) )
= One2 ) ) ) ) ).
% abstract_boolean_algebra_axioms_def
thf(fact_7780_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,One: A,Zero: A,Compl: A > A] :
( ! [X2: A,Y2: A,Z3: A] :
( ( Conj @ X2 @ ( Disj @ Y2 @ Z3 ) )
= ( Disj @ ( Conj @ X2 @ Y2 ) @ ( Conj @ X2 @ Z3 ) ) )
=> ( ! [X2: A,Y2: A,Z3: A] :
( ( Disj @ X2 @ ( Conj @ Y2 @ Z3 ) )
= ( Conj @ ( Disj @ X2 @ Y2 ) @ ( Disj @ X2 @ Z3 ) ) )
=> ( ! [X2: A] :
( ( Conj @ X2 @ One )
= X2 )
=> ( ! [X2: A] :
( ( Disj @ X2 @ Zero )
= X2 )
=> ( ! [X2: A] :
( ( Conj @ X2 @ ( Compl @ X2 ) )
= Zero )
=> ( ! [X2: A] :
( ( Disj @ X2 @ ( Compl @ X2 ) )
= One )
=> ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One ) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
thf(fact_7781_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: 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)
% Type constructors (836)
thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of,axiom,
code_term_of @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep,axiom,
typerep @ code_natural ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_2,axiom,
typerep @ code_integer ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_3,axiom,
code_term_of @ code_term ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_4,axiom,
typerep @ code_term ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_5,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( heap_ext @ A20 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_6,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( heap_ext @ A20 ) ) ) ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_7,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_8,axiom,
typerep @ product_unit ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_9,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( code_term_of @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_10,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( enum @ A21 ) )
=> ( enum @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_11,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( typerep @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_12,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( typerep @ ( old_node @ A20 @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_13,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_14,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( multiset @ A20 ) ) ) ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_15,axiom,
typerep @ assn ).
thf(tcon_String_Oliteral___Code__Evaluation_Oterm__of_16,axiom,
code_term_of @ literal ).
thf(tcon_String_Oliteral___Typerep_Otyperep_17,axiom,
typerep @ literal ).
thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_18,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( pred @ A20 ) ) ) ).
thf(tcon_Predicate_Opred___Typerep_Otyperep_19,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( pred @ A20 ) ) ) ).
thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_20,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( seq @ A20 ) ) ) ).
thf(tcon_Predicate_Oseq___Typerep_Otyperep_21,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( seq @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_22,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_23,axiom,
! [A20: $tType] :
( ( enum @ A20 )
=> ( enum @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_24,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( option @ A20 ) ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_25,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( filter @ A20 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_26,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( filter @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_27,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( code_term_of @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_28,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( enum @ A21 ) )
=> ( enum @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_29,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( typerep @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_30,axiom,
code_term_of @ char ).
thf(tcon_String_Ochar___Enum_Oenum_31,axiom,
enum @ char ).
thf(tcon_String_Ochar___Typerep_Otyperep_32,axiom,
typerep @ char ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_33,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_34,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( list @ A20 ) ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_35,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_36,axiom,
enum @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_37,axiom,
typerep @ $o ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_38,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_39,axiom,
! [A20: $tType] :
( ( enum @ A20 )
=> ( enum @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_40,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( set @ A20 ) ) ) ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_41,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_42,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_43,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_44,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_45,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_46,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_47,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Typerep_Otyperep_48,axiom,
typerep @ int ).
thf(tcon_itself___Code__Evaluation_Oterm__of_49,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( code_term_of @ ( itself @ A20 ) ) ) ).
thf(tcon_itself___Typerep_Otyperep_50,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( typerep @ ( itself @ A20 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_51,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( code_term_of @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Enum_Oenum_52,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( enum @ A21 ) )
=> ( enum @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep_53,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep @ A20 )
& ( typerep @ A21 ) )
=> ( typerep @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( condit1219197933456340205attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple592849572758109894attice @ A21 )
=> ( comple592849572758109894attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A20: $tType,A21: $tType] :
( ( comple489889107523837845lgebra @ A21 )
=> ( comple489889107523837845lgebra @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A20: $tType,A21: $tType] :
( ( ( cl_HOL_Oequal @ A20 )
& ( quickc3360725361186068524ustive @ A20 )
& ( quickc3360725361186068524ustive @ A21 ) )
=> ( quickc3360725361186068524ustive @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounde4967611905675639751up_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounde4346867609351753570nf_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple6319245703460814977attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A20: $tType,A21: $tType] :
( ( ( cl_HOL_Oequal @ A20 )
& ( quickc658316121487927005ustive @ A20 )
& ( quickc658316121487927005ustive @ A21 ) )
=> ( quickc658316121487927005ustive @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A20: $tType,A21: $tType] :
( ( boolea8198339166811842893lgebra @ A21 )
=> ( boolea8198339166811842893lgebra @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple9053668089753744459l_ccpo @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A20: $tType,A21: $tType] :
( ( ( code_term_of @ A20 )
& ( cl_HOL_Oequal @ A20 )
& ( quickcheck_random @ A21 ) )
=> ( quickcheck_random @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_sup @ A21 )
=> ( semilattice_sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_inf @ A21 )
=> ( semilattice_inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( distrib_lattice @ A21 )
=> ( distrib_lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A20: $tType,A21: $tType] :
( ( complete_Sup @ A21 )
=> ( complete_Sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A20: $tType,A21: $tType] :
( ( complete_Inf @ A21 )
=> ( complete_Inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A20: $tType,A21: $tType] :
( ( order_top @ A21 )
=> ( order_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( order_bot @ A21 )
=> ( order_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( countable @ A21 ) )
=> ( countable @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A20: $tType,A21: $tType] :
( ( preorder @ A21 )
=> ( preorder @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A20: $tType,A21: $tType] :
( ( lattice @ A21 )
=> ( lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A20: $tType,A21: $tType] :
( ( order @ A21 )
=> ( order @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A20: $tType,A21: $tType] :
( ( top @ A21 )
=> ( top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A20: $tType,A21: $tType] :
( ( ord @ A21 )
=> ( ord @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A20: $tType,A21: $tType] :
( ( bot @ A21 )
=> ( bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A20: $tType,A21: $tType] :
( ( uminus @ A21 )
=> ( uminus @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_sup @ A21 )
=> ( sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Oinf,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_inf @ A21 )
=> ( inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A20: $tType,A21: $tType] :
( ( minus @ A21 )
=> ( minus @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( cl_HOL_Oequal @ A21 ) )
=> ( cl_HOL_Oequal @ ( A20 > A21 ) ) ) ).
thf(tcon_itself___Quickcheck__Random_Orandom_54,axiom,
! [A20: $tType] :
( ( typerep @ A20 )
=> ( quickcheck_random @ ( itself @ A20 ) ) ) ).
thf(tcon_itself___HOL_Oequal_55,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( itself @ A20 ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_56,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_57,axiom,
quickc3360725361186068524ustive @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_58,axiom,
quickc658316121487927005ustive @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Quickcheck__Random_Orandom_59,axiom,
quickcheck_random @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_60,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_61,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_62,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OSup_63,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_64,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_65,axiom,
countable @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_66,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_67,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_68,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Orderings_Oord_69,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_70,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Lattices_Osup_71,axiom,
sup @ int ).
thf(tcon_Int_Oint___Lattices_Oinf_72,axiom,
inf @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_73,axiom,
minus @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___HOL_Oequal_74,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_75,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_76,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_77,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_78,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_79,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_80,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_81,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_82,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_83,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_84,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_85,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_86,axiom,
strict9044650504122735259up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
ordere1170586879665033532d_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_87,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_88,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_89,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_90,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_91,axiom,
quickc3360725361186068524ustive @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_92,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_93,axiom,
ordere8940638589300402666id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni5634975068530333245id_add @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_94,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_95,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_96,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_97,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_98,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_99,axiom,
quickc658316121487927005ustive @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_100,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_101,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_102,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_103,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_104,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_105,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_106,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_107,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_108,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_109,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_110,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_111,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_112,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_113,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_114,axiom,
quickcheck_random @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_115,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_116,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_117,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_118,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_119,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_120,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_121,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_122,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_123,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_124,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_125,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_126,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_127,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_128,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_129,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_130,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_131,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_132,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_133,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_134,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_135,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_136,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_137,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_138,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_139,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_140,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_141,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_142,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_143,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_144,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_145,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_146,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_147,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_148,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_149,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_150,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_151,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_152,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_153,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_154,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_155,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Lattices_Osup_156,axiom,
sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Oinf_157,axiom,
inf @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_158,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_159,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_160,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_161,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_162,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_163,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_164,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_165,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_166,axiom,
quickc3360725361186068524ustive @ num ).
thf(tcon_Num_Onum___Quickcheck__Random_Orandom_167,axiom,
quickcheck_random @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_168,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_169,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_170,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_171,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_172,axiom,
times @ num ).
thf(tcon_Num_Onum___HOL_Oequal_173,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_174,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_175,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_176,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_177,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_178,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_179,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_180,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_181,axiom,
quickc3360725361186068524ustive @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_182,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_183,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_184,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_185,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_186,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_187,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_188,axiom,
quickc658316121487927005ustive @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_189,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_190,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_191,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_192,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_193,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_194,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_195,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_196,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_197,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_198,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_199,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_200,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_201,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_202,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_203,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_204,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_205,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_206,axiom,
quickcheck_random @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_207,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_208,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_209,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_210,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_211,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_212,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_213,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_214,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_215,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_216,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_217,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_218,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_219,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_220,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_221,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_222,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_223,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_224,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_225,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_226,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_227,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_228,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_229,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_230,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_231,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_232,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_233,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_234,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_235,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_236,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_237,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_238,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_239,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_240,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_241,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_242,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_243,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_244,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_245,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_246,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_247,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_248,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_249,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_250,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_251,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_252,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_253,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Lattices_Osup_254,axiom,
sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Oinf_255,axiom,
inf @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_256,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_257,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_258,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_259,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_260,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_261,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_262,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_263,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_264,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_265,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_266,axiom,
! [A20: $tType] : ( condit1219197933456340205attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_267,axiom,
! [A20: $tType] : ( comple592849572758109894attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_268,axiom,
! [A20: $tType] : ( comple489889107523837845lgebra @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_269,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_270,axiom,
! [A20: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_271,axiom,
! [A20: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_272,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_273,axiom,
! [A20: $tType] :
( ( quickc658316121487927005ustive @ A20 )
=> ( quickc658316121487927005ustive @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_274,axiom,
! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_275,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_276,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_277,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_278,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_279,axiom,
! [A20: $tType] : ( semilattice_sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_280,axiom,
! [A20: $tType] : ( semilattice_inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_281,axiom,
! [A20: $tType] : ( distrib_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_282,axiom,
! [A20: $tType] : ( bounded_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_283,axiom,
! [A20: $tType] : ( complete_Sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_284,axiom,
! [A20: $tType] : ( complete_Inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_285,axiom,
! [A20: $tType] : ( order_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_286,axiom,
! [A20: $tType] : ( order_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_287,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( countable @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_288,axiom,
! [A20: $tType] : ( preorder @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_289,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_290,axiom,
! [A20: $tType] : ( lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_291,axiom,
! [A20: $tType] : ( order @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_292,axiom,
! [A20: $tType] : ( top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_293,axiom,
! [A20: $tType] : ( ord @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_294,axiom,
! [A20: $tType] : ( bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_295,axiom,
! [A20: $tType] : ( uminus @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_296,axiom,
! [A20: $tType] : ( sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Oinf_297,axiom,
! [A20: $tType] : ( inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_298,axiom,
! [A20: $tType] : ( minus @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_299,axiom,
! [A20: $tType] :
( ( cl_HOL_Oequal @ A20 )
=> ( cl_HOL_Oequal @ ( set @ A20 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_300,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_301,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_302,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_303,axiom,
quickc3360725361186068524ustive @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_304,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_305,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_306,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_307,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_308,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_309,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_310,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_311,axiom,
quickcheck_random @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_312,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_313,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_314,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_315,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_316,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_317,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_318,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_319,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_320,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_321,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_322,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_323,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_324,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_325,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_326,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_327,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_328,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_329,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_330,axiom,
sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oinf_331,axiom,
inf @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_332,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_333,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_334,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Quickcheck__Random_Orandom_335,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Countable_Ocountable_336,axiom,
! [A20: $tType] :
( ( countable @ A20 )
=> ( countable @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_337,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( list @ A20 ) ) ).
thf(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_338,axiom,
quickc3360725361186068524ustive @ char ).
thf(tcon_String_Ochar___Quickcheck__Random_Orandom_339,axiom,
quickcheck_random @ char ).
thf(tcon_String_Ochar___Countable_Ocountable_340,axiom,
countable @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_341,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___HOL_Oequal_342,axiom,
cl_HOL_Oequal @ char ).
thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_343,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickc3360725361186068524ustive @ A20 )
& ( quickc3360725361186068524ustive @ A21 ) )
=> ( quickc3360725361186068524ustive @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_344,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickcheck_random @ A20 )
& ( quickcheck_random @ A21 ) )
=> ( quickcheck_random @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_345,axiom,
! [A20: $tType,A21: $tType] :
( ( ( countable @ A20 )
& ( countable @ A21 ) )
=> ( countable @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_346,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_347,axiom,
! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A20 @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_348,axiom,
! [A20: $tType] : ( condit1219197933456340205attice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_349,axiom,
! [A20: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_350,axiom,
! [A20: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_351,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_352,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_353,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_354,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_355,axiom,
! [A20: $tType] : ( semilattice_sup @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_356,axiom,
! [A20: $tType] : ( semilattice_inf @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_357,axiom,
! [A20: $tType] : ( distrib_lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_358,axiom,
! [A20: $tType] : ( bounded_lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_359,axiom,
! [A20: $tType] : ( complete_Sup @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_360,axiom,
! [A20: $tType] : ( complete_Inf @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_361,axiom,
! [A20: $tType] : ( order_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_362,axiom,
! [A20: $tType] : ( order_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_363,axiom,
! [A20: $tType] : ( preorder @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_364,axiom,
! [A20: $tType] : ( lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_365,axiom,
! [A20: $tType] : ( order @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_366,axiom,
! [A20: $tType] : ( top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_367,axiom,
! [A20: $tType] : ( ord @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_368,axiom,
! [A20: $tType] : ( bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osup_369,axiom,
! [A20: $tType] : ( sup @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Oinf_370,axiom,
! [A20: $tType] : ( inf @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_371,axiom,
! [A20: $tType] :
( ( cl_HOL_Oequal @ A20 )
=> ( cl_HOL_Oequal @ ( filter @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_372,axiom,
! [A20: $tType] :
( ( comple5582772986160207858norder @ A20 )
=> ( condit6923001295902523014norder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_373,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_374,axiom,
! [A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_375,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_376,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_377,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounde4346867609351753570nf_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A20: $tType] :
( ( comple5582772986160207858norder @ A20 )
=> ( comple5582772986160207858norder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_378,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_379,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_380,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( bounded_lattice_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_381,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_382,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_383,axiom,
! [A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_384,axiom,
! [A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_385,axiom,
! [A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_386,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OSup_387,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( complete_Sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OInf_388,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( complete_Inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_389,axiom,
! [A20: $tType] :
( ( wellorder @ A20 )
=> ( wellorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_390,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_391,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_392,axiom,
! [A20: $tType] :
( ( countable @ A20 )
=> ( countable @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_393,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_394,axiom,
! [A20: $tType] :
( ( linorder @ A20 )
=> ( linorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_395,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_396,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_397,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_398,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_399,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_400,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osup_401,axiom,
! [A20: $tType] :
( ( sup @ A20 )
=> ( sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Oinf_402,axiom,
! [A20: $tType] :
( ( inf @ A20 )
=> ( inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_403,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( option @ A20 ) ) ).
thf(tcon_Predicate_Oseq___HOL_Oequal_404,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( seq @ A20 ) ) ).
thf(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_405,axiom,
! [A20: $tType] : ( condit1219197933456340205attice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_406,axiom,
! [A20: $tType] : ( comple592849572758109894attice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__boolean__algebra_407,axiom,
! [A20: $tType] : ( comple489889107523837845lgebra @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_408,axiom,
! [A20: $tType] : ( bounde4967611905675639751up_bot @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_409,axiom,
! [A20: $tType] : ( bounde4346867609351753570nf_top @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_410,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_411,axiom,
! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_412,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_413,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_414,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_415,axiom,
! [A20: $tType] : ( semilattice_sup @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_416,axiom,
! [A20: $tType] : ( semilattice_inf @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Odistrib__lattice_417,axiom,
! [A20: $tType] : ( distrib_lattice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_418,axiom,
! [A20: $tType] : ( bounded_lattice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OSup_419,axiom,
! [A20: $tType] : ( complete_Sup @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OInf_420,axiom,
! [A20: $tType] : ( complete_Inf @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__top_421,axiom,
! [A20: $tType] : ( order_top @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__bot_422,axiom,
! [A20: $tType] : ( order_bot @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Opreorder_423,axiom,
! [A20: $tType] : ( preorder @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Olattice_424,axiom,
! [A20: $tType] : ( lattice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder_425,axiom,
! [A20: $tType] : ( order @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Otop_426,axiom,
! [A20: $tType] : ( top @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oord_427,axiom,
! [A20: $tType] : ( ord @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Obot_428,axiom,
! [A20: $tType] : ( bot @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ouminus_429,axiom,
! [A20: $tType] : ( uminus @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osup_430,axiom,
! [A20: $tType] : ( sup @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Oinf_431,axiom,
! [A20: $tType] : ( inf @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ominus_432,axiom,
! [A20: $tType] : ( minus @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___HOL_Oequal_433,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( pred @ A20 ) ) ).
thf(tcon_String_Oliteral___Quickcheck__Random_Orandom_434,axiom,
quickcheck_random @ literal ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_435,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Countable_Ocountable_436,axiom,
countable @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_437,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_438,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_439,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_440,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_441,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_442,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___HOL_Oequal_443,axiom,
cl_HOL_Oequal @ literal ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_444,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_445,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_446,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_447,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_448,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_449,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_450,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_451,axiom,
dvd @ assn ).
thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_452,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_453,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_454,axiom,
! [A20: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_455,axiom,
! [A20: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_456,axiom,
! [A20: $tType] : ( cancel_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_457,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_458,axiom,
! [A20: $tType] : ( comm_monoid_diff @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_459,axiom,
! [A20: $tType] : ( ab_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_460,axiom,
! [A20: $tType] : ( comm_monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_461,axiom,
! [A20: $tType] : ( complete_Sup @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_462,axiom,
! [A20: $tType] : ( complete_Inf @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_463,axiom,
! [A20: $tType] : ( semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_464,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_465,axiom,
! [A20: $tType] : ( monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_466,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( order @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_467,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_468,axiom,
! [A20: $tType] : ( minus @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_469,axiom,
! [A20: $tType] : ( zero @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_470,axiom,
! [A20: $tType] :
( ( cl_HOL_Oequal @ A20 )
=> ( cl_HOL_Oequal @ ( multiset @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_471,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickc3360725361186068524ustive @ A20 )
& ( quickc3360725361186068524ustive @ A21 ) )
=> ( quickc3360725361186068524ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_472,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickc658316121487927005ustive @ A20 )
& ( quickc658316121487927005ustive @ A21 ) )
=> ( quickc658316121487927005ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_473,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickcheck_random @ A20 )
& ( quickcheck_random @ A21 ) )
=> ( quickcheck_random @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_474,axiom,
! [A20: $tType,A21: $tType] :
( ( ( countable @ A20 )
& ( countable @ A21 ) )
=> ( countable @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_475,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_476,axiom,
! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A20 @ A21 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_477,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_478,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_479,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_480,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_481,axiom,
quickc3360725361186068524ustive @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_482,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_483,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_484,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_485,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_486,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_487,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_488,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_489,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_490,axiom,
quickcheck_random @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_491,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_492,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_493,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_494,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_495,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_496,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_497,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_498,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_499,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_500,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_501,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_502,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_503,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_504,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_505,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_506,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_507,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_508,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_509,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_510,axiom,
sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oinf_511,axiom,
inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_512,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_513,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_514,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( heap_ext @ A20 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_515,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A20 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_516,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_517,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_518,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_519,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_520,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_521,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_522,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_523,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_524,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_525,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_526,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_527,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_528,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_529,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_530,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_531,axiom,
quickc3360725361186068524ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_532,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_533,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_534,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_535,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_536,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_537,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_538,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_539,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_540,axiom,
quickc658316121487927005ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_541,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_542,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_543,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_544,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_545,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_546,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_547,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_548,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_549,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_550,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_551,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_552,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_553,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_554,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_555,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_556,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_557,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_558,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_559,axiom,
quickcheck_random @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_560,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_561,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_562,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_563,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_564,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_565,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_566,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_567,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_568,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_569,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_570,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_571,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_572,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_573,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_574,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_575,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_576,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_577,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_578,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_579,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_580,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_581,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_582,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_583,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_584,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_585,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_586,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_587,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_588,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_589,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_590,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_591,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_592,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_593,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_594,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_595,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_596,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_597,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_598,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_599,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_600,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_601,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_602,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_603,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_604,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_605,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_606,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_607,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_608,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_609,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_610,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_611,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_612,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_613,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_614,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_615,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_616,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_617,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_618,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_619,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_620,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_621,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_622,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_623,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_624,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_625,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_626,axiom,
quickc3360725361186068524ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_627,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_628,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_629,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_630,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_631,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_632,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_633,axiom,
quickc658316121487927005ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_634,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_635,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_636,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_637,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_638,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_639,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_640,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_641,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_642,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_643,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_644,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_645,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_646,axiom,
quickcheck_random @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_647,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_648,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_649,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_650,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_651,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_652,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_653,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_654,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_655,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_656,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_657,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_658,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_659,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_660,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_661,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_662,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_663,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_664,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_665,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_666,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_667,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_668,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_669,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_670,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_671,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_672,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_673,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_674,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_675,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_676,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_677,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_678,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_679,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_680,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_681,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_682,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_683,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_684,axiom,
cl_HOL_Oequal @ code_natural ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X4: A] : ( P @ X4 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( x
= ( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ x @ H2 )
& ( rep_assn @ y @ H2 ) ) ) )
& ( x != y ) )
= ( ( x
= ( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ x @ H2 )
& ( rep_assn @ y @ H2 ) ) ) )
& ( y
!= ( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ y @ H2 )
& ( rep_assn @ x @ H2 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------