TPTP Problem File: ITP212^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP212^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00821_023885
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0024_Assertions_00821_023885 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 6668 (1715 unt; 891 typ; 0 def)
% Number of atoms : 18017 (6208 equ; 4 cnn)
% Maximal formula atoms : 48 ( 3 avg)
% Number of connectives : 133418 (1919 ~; 181 |;1152 &;121381 @)
% ( 0 <=>;8785 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 5366 (5366 >; 0 *; 0 +; 0 <<)
% Number of symbols : 885 ( 881 usr; 21 con; 0-9 aty)
% Number of variables : 21301 (1851 ^;18149 !; 329 ?;21301 :)
% ( 972 !>; 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 15:05:33.068
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
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_Typerep_Otyperep,type,
typerep: $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $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_Heap_Oarray,type,
array: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Heap_Oref,type,
ref: $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 (866)
thf(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[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_Heap_Oheap,type,
heap:
!>[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_Osup,type,
sup:
!>[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_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_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_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_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_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_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_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__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_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
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_Oentails,type,
entails: assn > assn > $o ).
thf(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( ( A > assn ) > assn ) ).
thf(sy_c_Assertions_Oin__range,type,
in_range: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Ois__pure__assn,type,
is_pure_assn: assn > $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_Oone__assn__raw__rel,type,
one_assn_raw_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( 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_Opure__assn,type,
pure_assn: $o > assn ).
thf(sy_c_Assertions_Opure__assn__raw,type,
pure_assn_raw:
!>[A: $tType,B: $tType] : ( $o > ( product_prod @ A @ ( set @ B ) ) > $o ) ).
thf(sy_c_Assertions_Opure__assn__raw__rel,type,
pure_assn_raw_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) > ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) > $o ) ).
thf(sy_c_Assertions_OrelH,type,
relH: ( set @ nat ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o ).
thf(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > assn ) ).
thf(sy_c_Assertions_Osnga__assn__raw,type,
snga_assn_raw:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
thf(sy_c_Assertions_Osnga__assn__raw__rel,type,
snga_assn_raw_rel:
!>[A: $tType] : ( ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
thf(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref @ A ) > A > assn ) ).
thf(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref @ A ) > A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
thf(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : ( ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $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_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_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_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__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_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__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_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_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_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
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_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_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_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_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_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $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_Countable_Onth__item__rel,type,
nth_item_rel: nat > nat > $o ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ 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_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_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_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_Ofinite,type,
finite_finite:
!>[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__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_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
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__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,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > 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_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( ( array @ A ) > nat ) ).
thf(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ( ref @ A ) > nat ) ).
thf(sy_c_Heap_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > ( array @ A ) ) ).
thf(sy_c_Heap_Oarray_Oset__array,type,
set_array:
!>[A: $tType] : ( ( array @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
thf(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
thf(sy_c_Heap_Oheap_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > nat ) ).
thf(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ( ref @ A ) ) ).
thf(sy_c_Heap_Oref_Oset__ref,type,
set_ref:
!>[A: $tType] : ( ( ref @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
heap_Time_Heap_lub:
!>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
thf(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
heap_Time_timeFrame:
!>[A: $tType] : ( nat > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
heap_T5500966940807335491me_rel:
!>[A: $tType] : ( ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Oureturn,type,
heap_Time_ureturn:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
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_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
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_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__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__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_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_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ 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_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__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ 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_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_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__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_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_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__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_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_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_Osuccessively,type,
successively:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
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_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > 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__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ 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_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__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_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_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_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_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_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_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_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_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_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_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_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
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_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_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_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_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_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_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_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_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__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_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_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > A > A > $o ) ).
thf(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
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_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
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_Onot__pred,type,
not_pred: ( pred @ product_unit ) > ( pred @ product_unit ) ).
thf(sy_c_Predicate_Opred_OPred,type,
pred2:
!>[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_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__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_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > 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_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_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_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_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_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Ref__Time_Oalloc,type,
ref_alloc:
!>[A: $tType] : ( A > ( heap_ext @ product_unit ) > ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Ref__Time_Ochange,type,
ref_change:
!>[A: $tType] : ( ( A > A ) > ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
thf(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Opresent,type,
ref_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > $o ) ).
thf(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > ( heap_Time_Heap @ ( ref @ A ) ) ) ).
thf(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
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_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_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind2:
!>[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_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_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_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_Syntax__Match_Oac__operator,type,
syntax_ac_operator:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Syntax__Match_Osyntax__nomatch,type,
syntax2379306206330768139omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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_Oacc,type,
acc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
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_Wfrec_Owfrec__rel,type,
wfrec_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > A > B > $o ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
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_Osuc__Union__closed,type,
pred_s596693808085603175closed:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ ( set @ A ) ) ) ).
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_A,type,
a: assn ).
thf(sy_v_B,type,
b: assn ).
thf(sy_v_C,type,
c: assn ).
% Relevant facts (4970)
thf(fact_0_ent__iffI,axiom,
! [A3: assn,B2: assn] :
( ( entails @ A3 @ B2 )
=> ( ( entails @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% ent_iffI
thf(fact_1_ent__refl,axiom,
! [P: assn] : ( entails @ P @ P ) ).
% ent_refl
thf(fact_2_ent__disjE,axiom,
! [A3: assn,C2: assn,B2: assn] :
( ( entails @ A3 @ C2 )
=> ( ( entails @ B2 @ C2 )
=> ( entails @ ( sup_sup @ assn @ A3 @ B2 ) @ C2 ) ) ) ).
% ent_disjE
thf(fact_3_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ Q )
=> ( ( entails @ Q @ R )
=> ( entails @ P @ R ) ) ) ).
% ent_trans
thf(fact_4_ent__disjI1,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R )
=> ( entails @ P @ R ) ) ).
% ent_disjI1
thf(fact_5_ent__disjI2,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R )
=> ( entails @ Q @ R ) ) ).
% ent_disjI2
thf(fact_6_ent__star__mono,axiom,
! [P: assn,P2: assn,Q: assn,Q2: assn] :
( ( entails @ P @ P2 )
=> ( ( entails @ Q @ Q2 )
=> ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P2 @ Q2 ) ) ) ) ).
% ent_star_mono
thf(fact_7_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P3: assn,Q3: assn] : ( times_times @ assn @ Q3 @ P3 ) ) ) ).
% assn_times_comm
thf(fact_8_assn__times__assoc,axiom,
! [P: assn,Q: assn,R: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ R )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q @ R ) ) ) ).
% assn_times_assoc
thf(fact_9_ent__disjI1__direct,axiom,
! [A3: assn,B2: assn] : ( entails @ A3 @ ( sup_sup @ assn @ A3 @ B2 ) ) ).
% ent_disjI1_direct
thf(fact_10_ent__disjI2__direct,axiom,
! [B2: assn,A3: assn] : ( entails @ B2 @ ( sup_sup @ assn @ A3 @ B2 ) ) ).
% ent_disjI2_direct
thf(fact_11_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_12_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_13_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_14_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_15_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ B3 )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_16_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_17_ent__mp,axiom,
! [P: assn,Q: assn] : ( entails @ ( times_times @ assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).
% ent_mp
thf(fact_18_ent__wandI,axiom,
! [Q: assn,P: assn,R: assn] :
( ( entails @ ( times_times @ assn @ Q @ P ) @ R )
=> ( entails @ P @ ( wand_assn @ Q @ R ) ) ) ).
% ent_wandI
thf(fact_19_ent__pure__pre__iff,axiom,
! [P: assn,B3: $o,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ Q )
= ( B3
=> ( entails @ P @ Q ) ) ) ).
% ent_pure_pre_iff
thf(fact_20_is__pure__assn__starI,axiom,
! [A4: assn,B3: assn] :
( ( is_pure_assn @ A4 )
=> ( ( is_pure_assn @ B3 )
=> ( is_pure_assn @ ( times_times @ assn @ A4 @ B3 ) ) ) ) ).
% is_pure_assn_starI
thf(fact_21_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_22_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_23_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q ) )
= ( P = Q ) ) ).
% pure_assn_eq_conv
thf(fact_24_merge__pure__star,axiom,
! [A4: $o,B3: $o] :
( ( times_times @ assn @ ( pure_assn @ A4 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A4
& B3 ) ) ) ).
% merge_pure_star
thf(fact_25_merge__pure__or,axiom,
! [A4: $o,B3: $o] :
( ( sup_sup @ assn @ ( pure_assn @ A4 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A4
| B3 ) ) ) ).
% merge_pure_or
thf(fact_26_is__pure__assn__pure,axiom,
! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% is_pure_assn_pure
thf(fact_27_is__pure__assnE,axiom,
! [A4: assn] :
( ( is_pure_assn @ A4 )
=> ~ ! [P4: $o] :
( A4
!= ( pure_assn @ P4 ) ) ) ).
% is_pure_assnE
thf(fact_28_is__pure__assn__def,axiom,
( is_pure_assn
= ( ^ [A5: assn] :
? [P3: $o] :
( A5
= ( pure_assn @ P3 ) ) ) ) ).
% is_pure_assn_def
thf(fact_29_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_30_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_31_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A4 @ C3 ) )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_32_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_33_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B4: A] : ( sup_sup @ A @ B4 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_34_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_35_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ C3 )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_36_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_37_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_38_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B3: $o] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B3 ) ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> B3 )
& ( entails @ P @ Q ) ) ) ).
% ent_pure_post_iff
thf(fact_39_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A4: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A4 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_40_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,K: A,A4: A,B3: A] :
( ( A3
= ( sup_sup @ A @ K @ A4 ) )
=> ( ( sup_sup @ A @ A3 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_41_ent__pure__pre__iff__sng,axiom,
! [B3: $o,Q: assn] :
( ( entails @ ( pure_assn @ B3 ) @ Q )
= ( B3
=> ( entails @ ( one_one @ assn ) @ Q ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_42_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( times_times @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_43_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_44_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G2 @ X3 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( times_times @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ).
% mult.right_commute
thf(fact_50_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% mult.right_assoc
thf(fact_51_mod__pure__star__dist,axiom,
! [P: assn,B3: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ H2 )
= ( ( rep_assn @ P @ H2 )
& B3 ) ) ).
% mod_pure_star_dist
thf(fact_52_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_53_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_54_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_55_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.right_neutral
thf(fact_56_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% mult_1
thf(fact_57_mod__or__dist,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ H2 )
= ( ( rep_assn @ P @ H2 )
| ( rep_assn @ Q @ H2 ) ) ) ).
% mod_or_dist
thf(fact_58_ent__pure__post__iff__sng,axiom,
! [P: assn,B3: $o] :
( ( entails @ P @ ( pure_assn @ B3 ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> B3 )
& ( entails @ P @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_59_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_60_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_61_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.comm_neutral
thf(fact_62_ent__fwd,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q: assn] :
( ( rep_assn @ P @ H2 )
=> ( ( entails @ P @ Q )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% ent_fwd
thf(fact_63_entailsD,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P @ Q )
=> ( ( rep_assn @ P @ H2 )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% entailsD
thf(fact_64_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H3 )
=> ( rep_assn @ Q @ H3 ) )
=> ( entails @ P @ Q ) ) ).
% entailsI
thf(fact_65_entails__def,axiom,
( entails
= ( ^ [P3: assn,Q3: assn] :
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H )
=> ( rep_assn @ Q3 @ H ) ) ) ) ).
% entails_def
thf(fact_66_mod__starD,axiom,
! [A3: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H2 )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A3 @ H1 )
& ( rep_assn @ B2 @ H22 ) ) ) ).
% mod_starD
thf(fact_67_mod__starE,axiom,
! [A4: assn,B3: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A4 @ B3 ) @ H2 )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A4 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B3 @ H_2 ) ) ) ).
% mod_starE
thf(fact_68_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_69_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn @ ( one_one @ assn ) ).
% is_pure_assn_basic_simps(2)
thf(fact_70_mult_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [X: A,Y: A,A4: A,B3: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ B3 )
= ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% mult.safe_commute
thf(fact_71_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_72_ent__false__iff,axiom,
! [P: assn] :
( ( entails @ P @ ( bot_bot @ assn ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H ) ) ) ).
% ent_false_iff
thf(fact_73_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.comm_monoid_axioms
thf(fact_74_mod__star__trueE,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H4 ) ) ).
% mod_star_trueE
thf(fact_75_mod__star__trueI,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 ) ) ).
% mod_star_trueI
thf(fact_76_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_77_wand__assn__def,axiom,
( wand_assn
= ( ^ [P3: assn,Q3: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).
% wand_assn_def
thf(fact_78_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_list_set_axioms
thf(fact_79_syntax__nomatch__def,axiom,
! [B: $tType,A: $tType] :
( ( syntax2379306206330768139omatch @ A @ B )
= ( ^ [Pat: A,Obj: B] : $true ) ) ).
% syntax_nomatch_def
thf(fact_80_mult_Oac__operator__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( syntax_ac_operator @ A @ ( times_times @ A ) ) ) ).
% mult.ac_operator_axioms
thf(fact_81_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.comm_monoid_list_axioms
thf(fact_82_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ ( bot_bot @ A ) )
= A4 ) ) ).
% sup_bot.right_neutral
thf(fact_83_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A4 @ B3 ) )
= ( ( A4
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_84_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A4 )
= A4 ) ) ).
% sup_bot.left_neutral
thf(fact_85_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A,B3: A] :
( ( ( sup_sup @ A @ A4 @ B3 )
= ( bot_bot @ A ) )
= ( ( A4
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_86_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_87_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_88_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_89_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_90_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_91_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_92_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_93_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_94_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_95_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_96_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_97_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_98_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_99_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_100_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_101_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_102_ac__operator_Osafe__commute,axiom,
! [A: $tType,F2: A > A > A,X: A,Y: A,A4: A,B3: A] :
( ( syntax_ac_operator @ A @ F2 )
=> ( ( syntax7388354845996824322omatch @ A @ A @ ( F2 @ X @ Y ) @ A4 )
=> ( ( F2 @ A4 @ B3 )
= ( F2 @ B3 @ A4 ) ) ) ) ).
% ac_operator.safe_commute
thf(fact_103_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ A4 @ Z2 )
= A4 ) ) ).
% comm_monoid.comm_neutral
thf(fact_104_monoid_Oright__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ A4 @ Z2 )
= A4 ) ) ).
% monoid.right_neutral
thf(fact_105_monoid_Oleft__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ Z2 @ A4 )
= A4 ) ) ).
% monoid.left_neutral
thf(fact_106_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_107_comm__monoid__list__set_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups4802862169904069756st_set @ A @ F2 @ Z2 )
=> ( groups1828464146339083142d_list @ A @ F2 @ Z2 ) ) ).
% comm_monoid_list_set.axioms(1)
thf(fact_108_ac__operator_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A,B5: A,C4: A] :
( ( F2 @ ( F2 @ A6 @ B5 ) @ C4 )
= ( F2 @ A6 @ ( F2 @ B5 @ C4 ) ) )
=> ( ! [A6: A,B5: A] :
( ( F2 @ A6 @ B5 )
= ( F2 @ B5 @ A6 ) )
=> ( syntax_ac_operator @ A @ F2 ) ) ) ).
% ac_operator.intro
thf(fact_109_ac__operator_Ocommute,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A] :
( ( syntax_ac_operator @ A @ F2 )
=> ( ( F2 @ A4 @ B3 )
= ( F2 @ B3 @ A4 ) ) ) ).
% ac_operator.commute
thf(fact_110_ac__operator_Oleft__assoc,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A,C3: A] :
( ( syntax_ac_operator @ A @ F2 )
=> ( ( F2 @ A4 @ ( F2 @ B3 @ C3 ) )
= ( F2 @ ( F2 @ A4 @ B3 ) @ C3 ) ) ) ).
% ac_operator.left_assoc
thf(fact_111_ac__operator_Oright__assoc,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A,C3: A] :
( ( syntax_ac_operator @ A @ F2 )
=> ( ( F2 @ ( F2 @ A4 @ B3 ) @ C3 )
= ( F2 @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ).
% ac_operator.right_assoc
thf(fact_112_ac__operator_Oleft__commute,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A,C3: A] :
( ( syntax_ac_operator @ A @ F2 )
=> ( ( F2 @ A4 @ ( F2 @ B3 @ C3 ) )
= ( F2 @ B3 @ ( F2 @ A4 @ C3 ) ) ) ) ).
% ac_operator.left_commute
thf(fact_113_ac__operator_Oright__commute,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A,C3: A] :
( ( syntax_ac_operator @ A @ F2 )
=> ( ( F2 @ ( F2 @ A4 @ B3 ) @ C3 )
= ( F2 @ ( F2 @ A4 @ C3 ) @ B3 ) ) ) ).
% ac_operator.right_commute
thf(fact_114_ac__operator__def,axiom,
! [A: $tType] :
( ( syntax_ac_operator @ A )
= ( ^ [F: A > A > A] :
( ! [A5: A,B4: A,C5: A] :
( ( F @ ( F @ A5 @ B4 ) @ C5 )
= ( F @ A5 @ ( F @ B4 @ C5 ) ) )
& ! [A5: A,B4: A] :
( ( F @ A5 @ B4 )
= ( F @ B4 @ A5 ) ) ) ) ) ).
% ac_operator_def
thf(fact_115_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.comm_monoid_axioms
thf(fact_116_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_117_syntax__fo__nomatch__def,axiom,
! [B: $tType,A: $tType] :
( ( syntax7388354845996824322omatch @ A @ B )
= ( ^ [Pat: A,Obj: B] : $true ) ) ).
% syntax_fo_nomatch_def
thf(fact_118_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_119_mod__false,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H2 ) ).
% mod_false
thf(fact_120_ent__true,axiom,
! [P: assn] : ( entails @ P @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_121_ent__false,axiom,
! [P: assn] : ( entails @ ( bot_bot @ assn ) @ P ) ).
% ent_false
thf(fact_122_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn @ ( bot_bot @ assn ) ).
% is_pure_assn_basic_simps(1)
thf(fact_123_Abs__assn__eqI_I2_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H3 )
= ( rep_assn @ Pr @ H3 ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_124_Abs__assn__eqI_I1_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H3 )
= ( rep_assn @ Pr @ H3 ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_125_sngr__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P5: ref @ A,X: A,Y: A] :
( ( times_times @ assn @ ( sngr_assn @ A @ P5 @ X ) @ ( sngr_assn @ A @ P5 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% sngr_same_false
thf(fact_126_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P5: array @ A,X: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P5 @ X ) @ ( snga_assn @ A @ P5 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_127_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_128_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_129_times__assn__def,axiom,
( ( times_times @ assn )
= ( ^ [P3: assn,Q3: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).
% times_assn_def
thf(fact_130_one__assn__def,axiom,
( ( one_one @ assn )
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_131_pure__assn__def,axiom,
( pure_assn
= ( ^ [B4: $o] : ( abs_assn @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B4 ) ) ) ) ).
% pure_assn_def
thf(fact_132_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_133_comm__monoid__list__def,axiom,
! [A: $tType] :
( ( groups1828464146339083142d_list @ A )
= ( ^ [F: A > A > A,Z3: A] :
( ( comm_monoid @ A @ F @ Z3 )
& ( groups_monoid_list @ A @ F @ Z3 ) ) ) ) ).
% comm_monoid_list_def
thf(fact_134_mod__true,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( top_top @ assn ) @ H2 )
= ( in_range @ H2 ) ) ).
% mod_true
thf(fact_135_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_136_prod__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.monoid_list_axioms
thf(fact_137_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F: A > A > A,Z3: A] :
( ! [A5: A] :
( ( F @ Z3 @ A5 )
= A5 )
& ! [A5: A] :
( ( F @ A5 @ Z3 )
= A5 ) ) ) ) ).
% monoid_axioms_def
thf(fact_138_monoid__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ! [A6: A] :
( ( F2 @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F2 @ A6 @ Z2 )
= A6 )
=> ( monoid_axioms @ A @ F2 @ Z2 ) ) ) ).
% monoid_axioms.intro
thf(fact_139_models__in__range,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> ( in_range @ H2 ) ) ).
% models_in_range
thf(fact_140_monoid__list__def,axiom,
! [A: $tType] :
( ( groups_monoid_list @ A )
= ( monoid @ A ) ) ).
% monoid_list_def
thf(fact_141_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_142_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_143_comm__monoid__list_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups1828464146339083142d_list @ A @ F2 @ Z2 )
=> ( groups_monoid_list @ A @ F2 @ Z2 ) ) ).
% comm_monoid_list.axioms(2)
thf(fact_144_top__assn__def,axiom,
( ( top_top @ assn )
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_145_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_146_sngr__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( sngr_assn @ A )
= ( ^ [R2: ref @ A,X2: A] : ( abs_assn @ ( sngr_assn_raw @ A @ R2 @ X2 ) ) ) ) ) ).
% sngr_assn_def
thf(fact_147_snga__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( snga_assn @ A )
= ( ^ [R2: array @ A,A5: list @ A] : ( abs_assn @ ( snga_assn_raw @ A @ R2 @ A5 ) ) ) ) ) ).
% snga_assn_def
thf(fact_148_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_149_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F: A > A > A,Z3: A] :
( ( semigroup @ A @ F )
& ( monoid_axioms @ A @ F @ Z3 ) ) ) ) ).
% monoid_def
thf(fact_150_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P5: ref @ A,X: A,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( sngr_assn @ A @ P5 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(3)
thf(fact_151_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q4: array @ B,Y: list @ B,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q4 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_152_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_153_pure__assn__proper,axiom,
! [B3: $o] : ( proper @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B3 ) ) ).
% pure_assn_proper
thf(fact_154_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_155_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
| ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(7)
thf(fact_156_mod__h__bot__iff_I1_J,axiom,
! [B3: $o,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= B3 ) ).
% mod_h_bot_iff(1)
thf(fact_157_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_158_in__range__dist__union,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As @ As2 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) ) ) ) ).
% in_range_dist_union
thf(fact_159_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_160_snga__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: array @ A,X: list @ A] : ( proper @ ( snga_assn_raw @ A @ R3 @ X ) ) ) ).
% snga_assn_proper
thf(fact_161_sngr__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,X: A] : ( proper @ ( sngr_assn_raw @ A @ R3 @ X ) ) ) ).
% sngr_assn_proper
thf(fact_162_wand__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] : ( proper @ ( wand_raw @ P @ Q ) ) ).
% wand_proper
thf(fact_163_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_164_times__assn__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q )
=> ( proper @ ( times_assn_raw @ P @ Q ) ) ) ) ).
% times_assn_proper
thf(fact_165_in__range__empty,axiom,
! [H2: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% in_range_empty
thf(fact_166_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_167_semigroup_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A,B5: A,C4: A] :
( ( F2 @ ( F2 @ A6 @ B5 ) @ C4 )
= ( F2 @ A6 @ ( F2 @ B5 @ C4 ) ) )
=> ( semigroup @ A @ F2 ) ) ).
% semigroup.intro
thf(fact_168_semigroup_Oassoc,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A,C3: A] :
( ( semigroup @ A @ F2 )
=> ( ( F2 @ ( F2 @ A4 @ B3 ) @ C3 )
= ( F2 @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ).
% semigroup.assoc
thf(fact_169_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ).
% one_assn_raw.cases
thf(fact_170_properD1,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% properD1
thf(fact_171_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ! [A6: A] :
( ( F2 @ A6 @ Z2 )
= A6 )
=> ( comm_monoid_axioms @ A @ F2 @ Z2 ) ) ).
% comm_monoid_axioms.intro
thf(fact_172_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F: A > A > A,Z3: A] :
! [A5: A] :
( ( F @ A5 @ Z3 )
= A5 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_173_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_174_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_175_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 ) )
=> ~ ! [X3: assn] :
( Y
!= ( rep_assn @ X3 ) ) ) ).
% Rep_assn_cases
thf(fact_176_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 ) )
=> ( ! [X3: assn] : ( P @ ( rep_assn @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_assn_induct
thf(fact_177_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_178_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_179_pure__assn__raw_Osimps,axiom,
! [A: $tType,B: $tType,B3: $o,H2: A,As: set @ B] :
( ( pure_assn_raw @ A @ B @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
= ( ( As
= ( bot_bot @ ( set @ B ) ) )
& B3 ) ) ).
% pure_assn_raw.simps
thf(fact_180_pure__assn__raw_Oelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X @ Xa )
= Y )
=> ~ ! [H3: A,As3: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) )
=> ( Y
= ( ~ ( ( As3
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_181_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X @ Xa )
=> ~ ! [H3: A,As3: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) )
=> ~ ( ( As3
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_182_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
=> ~ ! [H3: A,As3: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) )
=> ( ( As3
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_183_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( X
= ( abs_assn @ Y3 ) )
=> ~ ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y3 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ) ) ).
% Abs_assn_cases
thf(fact_184_Abs__assn__induct,axiom,
! [P: assn > $o,X: assn] :
( ! [Y3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y3 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( P @ ( abs_assn @ Y3 ) ) )
=> ( P @ X ) ) ).
% Abs_assn_induct
thf(fact_185_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_186_one__assn__raw_Osimps,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% one_assn_raw.simps
thf(fact_187_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( Y
= ( As3
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_188_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( As3
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_189_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( As3
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_190_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_191_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_192_mod__h__bot__iff_I2_J,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_193_mod__emp__simp,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp_simp
thf(fact_194_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_195_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_196_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_197_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_198_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_199_Un__empty,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_200_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_201_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A7: A,B6: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B6 ) )
= ( ( A4 = A7 )
& ( B3 = B6 ) ) ) ).
% old.prod.inject
thf(fact_202_mod__star__conv,axiom,
! [A3: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H2 )
= ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
( ( H2
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( rep_assn @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
& ( rep_assn @ B2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_203_star__assnI,axiom,
! [P: assn,H2: heap_ext @ product_unit,As: set @ nat,Q: assn,As2: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As @ As2 ) ) ) ) ) ) ).
% star_assnI
thf(fact_204_proper__iff,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( relH @ As @ H2 @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) ) ) ) ) ) ).
% proper_iff
thf(fact_205_proper__def,axiom,
( proper
= ( ^ [P3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
! [H: heap_ext @ product_unit,H6: heap_ext @ product_unit,As4: set @ nat] :
( ( ( P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) ) )
& ( ( ( P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ( relH @ As4 @ H @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) ) )
=> ( P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) ) ) ) ) ) ).
% proper_def
thf(fact_206_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A] :
( ( inf_inf @ A @ A4 @ A4 )
= A4 ) ) ).
% inf.idem
thf(fact_207_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_208_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( inf_inf @ A @ A4 @ ( inf_inf @ A @ A4 @ B3 ) )
= ( inf_inf @ A @ A4 @ B3 ) ) ) ).
% inf.left_idem
thf(fact_209_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_210_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A4 @ B3 ) @ B3 )
= ( inf_inf @ A @ A4 @ B3 ) ) ) ).
% inf.right_idem
thf(fact_211_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_212_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_213_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_214_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_215_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_216_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_217_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_218_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_219_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_220_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_221_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A,B3: A] :
( ( ( inf_inf @ A @ A4 @ B3 )
= ( top_top @ A ) )
= ( ( A4
= ( top_top @ A ) )
& ( B3
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_222_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A4 )
= A4 ) ) ).
% inf_top.left_neutral
thf(fact_223_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A,B3: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A4 @ B3 ) )
= ( ( A4
= ( top_top @ A ) )
& ( B3
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_224_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A] :
( ( inf_inf @ A @ A4 @ ( top_top @ A ) )
= A4 ) ) ).
% inf_top.right_neutral
thf(fact_225_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_226_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_227_relH__dist__union,axiom,
! [As: set @ nat,As2: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ ( sup_sup @ ( set @ nat ) @ As @ As2 ) @ H2 @ H5 )
= ( ( relH @ As @ H2 @ H5 )
& ( relH @ As2 @ H2 @ H5 ) ) ) ).
% relH_dist_union
thf(fact_228_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_229_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_230_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_231_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y2: A] : ( inf_inf @ A @ Y2 @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_232_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C3 )
= ( inf_inf @ A @ A4 @ ( inf_inf @ A @ B3 @ C3 ) ) ) ) ).
% inf.assoc
thf(fact_233_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_234_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B4: A] : ( inf_inf @ A @ B4 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_235_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y2: A] : ( inf_inf @ A @ Y2 @ X2 ) ) ) ) ).
% inf_commute
thf(fact_236_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A4 @ C3 ) )
= ( inf_inf @ A @ A4 @ ( inf_inf @ A @ B3 @ C3 ) ) ) ) ).
% inf.left_commute
thf(fact_237_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_238_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_239_relH__trans,axiom,
! [As: set @ nat,H12: heap_ext @ product_unit,H23: heap_ext @ product_unit,H32: heap_ext @ product_unit] :
( ( relH @ As @ H12 @ H23 )
=> ( ( relH @ As @ H23 @ H32 )
=> ( relH @ As @ H12 @ H32 ) ) ) ).
% relH_trans
thf(fact_240_relH__sym,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ As @ H2 @ H5 )
=> ( relH @ As @ H5 @ H2 ) ) ).
% relH_sym
thf(fact_241_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,K: A,A4: A,B3: A] :
( ( A3
= ( inf_inf @ A @ K @ A4 ) )
=> ( ( inf_inf @ A @ A3 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_242_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,K: A,B3: A,A4: A] :
( ( B2
= ( inf_inf @ A @ K @ B3 ) )
=> ( ( inf_inf @ A @ A4 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_243_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 ) ) )] :
~ ! [P4: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q5: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As3: 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 ) ) ) @ P4 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ).
% times_assn_raw.cases
thf(fact_244_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R4: ref @ A,X3: A,H3: heap_ext @ product_unit,As3: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R4 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_245_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R4: array @ A,X3: list @ A,H3: heap_ext @ product_unit,As3: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R4 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_246_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_247_Int__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_248_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_249_disjoint__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_250_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_251_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_252_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_253_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_254_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_255_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_256_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z4 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( inf_inf @ A @ X3 @ Z4 ) ) )
=> ( ( 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_257_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z4 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( sup_sup @ A @ X3 @ Z4 ) ) )
=> ( ( 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_258_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_259_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_260_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_261_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_262_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B5: $o,H3: A,As3: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B5 @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) ) ) ).
% pure_assn_raw.cases
thf(fact_263_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.semigroup_axioms
thf(fact_264_inf__top_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( comm_monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.comm_monoid_axioms
thf(fact_265_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.monoid_axioms
thf(fact_266_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semilattice_neutr @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.semilattice_neutr_axioms
thf(fact_267_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H6: heap_ext @ product_unit,As5: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As5 ) ) )
=> ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As5 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_268_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 )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
& ! [H6: heap_ext @ product_unit,As5: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As3 @ H3 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As3 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As5 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As3 @ As5 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_269_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 )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
& ! [H7: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As3 @ H3 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As3 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As3 @ As6 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_270_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 )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
& ! [H4: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As3 @ H3 @ H4 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( sup_sup @ ( set @ nat ) @ As3 @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_271_mod__relH,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit,P: assn] :
( ( relH @ As @ H2 @ H5 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) ) ) ) ).
% mod_relH
thf(fact_272_relH__refl,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( relH @ As @ H2 @ H2 ) ) ).
% relH_refl
thf(fact_273_relH__in__rangeI_I1_J,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ As @ H2 @ H5 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ).
% relH_in_rangeI(1)
thf(fact_274_relH__in__rangeI_I2_J,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ As @ H2 @ H5 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) ) ) ).
% relH_in_rangeI(2)
thf(fact_275_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A4: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A4 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A4 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A4 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A4 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_276_prod__induct7,axiom,
! [G3: $tType,F3: $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 @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A6: A,B5: B,C4: C,D2: D,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_277_prod__induct6,axiom,
! [F3: $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 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A6: A,B5: B,C4: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_278_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 ) ) )] :
( ! [A6: A,B5: B,C4: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_279_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 ) )] :
( ! [A6: A,B5: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_280_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 )] :
( ! [A6: A,B5: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_281_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A6: A,B5: B,C4: C,D2: D,E2: E,F4: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_282_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A6: A,B5: B,C4: C,D2: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_283_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B5: B,C4: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_284_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_285_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C4 ) ) ) ).
% prod_cases3
thf(fact_286_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A7: A,B6: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B6 ) )
=> ~ ( ( A4 = A7 )
=> ( B3 != B6 ) ) ) ).
% Pair_inject
thf(fact_287_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P5: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_288_surj__pair,axiom,
! [A: $tType,B: $tType,P5: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P5
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_289_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_290_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_291_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_292_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_293_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_294_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_295_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_296_equals0D,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A3 ) ) ).
% equals0D
thf(fact_297_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_298_wand__assnI,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat,Q: assn,R: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ! [H4: heap_ext @ product_unit,As7: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As @ H2 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As7 ) )
=> ( rep_assn @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% wand_assnI
thf(fact_299_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
& ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_300_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 )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( Y
= ( ~ ? [As1: set @ nat,As22: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_301_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 )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ~ ? [As12: set @ nat,As23: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_302_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 )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ? [As13: set @ nat,As24: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_303_properI,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [As3: set @ nat,H3: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) )
=> ( ! [As3: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( relH @ As3 @ H3 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_304_properD2,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat,H5: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( relH @ As @ H2 @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) ) ) ) ) ) ).
% properD2
thf(fact_305_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( F1 @ A4 @ B3 ) ) ).
% old.prod.rec
thf(fact_306_disjointI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ~ ( member @ A @ X3 @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_307_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 ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( 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 ) @ H3 @ As3 ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
& ! [H4: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As3 @ H3 @ H4 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( sup_sup @ ( set @ nat ) @ As3 @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_308_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 ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( 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 ) @ H3 @ As3 ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
& ! [H7: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As3 @ H3 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As3 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As3 @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_309_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 ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
& ! [H6: heap_ext @ product_unit,As5: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As3 @ H3 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As3 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As5 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As3 @ As5 ) ) ) ) ) )
=> ~ ( 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 ) @ H3 @ As3 ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_310_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty2 @ A )
= ( ^ [A8: set @ A] :
( A8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_311_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A4: B,B3: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( C3 @ A4 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_312_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 ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( 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 ) @ H3 @ As3 ) ) ) )
=> ? [As13: set @ nat,As24: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_313_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 ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( 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 ) @ H3 @ As3 ) ) ) )
=> ~ ? [As12: set @ nat,As23: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_314_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 ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( Y
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As22 ) ) ) ) )
=> ~ ( 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 ) @ H3 @ As3 ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_315_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_316_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_317_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_318_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B3 ) )
= ( A4 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_319_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
= A4 ) ) ).
% add.inverse_inverse
thf(fact_320_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A8: A > B,X2: A] : ( uminus_uminus @ B @ ( A8 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_321_merge__pure__and,axiom,
! [A4: $o,B3: $o] :
( ( inf_inf @ assn @ ( pure_assn @ A4 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A4
& B3 ) ) ) ).
% merge_pure_and
thf(fact_322_Compl__disjoint2,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_323_Compl__disjoint,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_324_mod__not__dist,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( uminus_uminus @ assn @ P ) @ H2 )
= ( ( in_range @ H2 )
& ~ ( rep_assn @ P @ H2 ) ) ) ).
% mod_not_dist
thf(fact_325_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_326_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(6)
thf(fact_339_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A4 ) ) ) ).
% minus_equation_iff
thf(fact_340_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% equation_minus_iff
thf(fact_341_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A8: A > B,X2: A] : ( uminus_uminus @ B @ ( A8 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_342_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_343_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_344_mod__and__dist,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ H2 )
= ( ( rep_assn @ P @ H2 )
& ( rep_assn @ Q @ H2 ) ) ) ).
% mod_and_dist
thf(fact_345_ent__conjE2,axiom,
! [B2: assn,C2: assn,A3: assn] :
( ( entails @ B2 @ C2 )
=> ( entails @ ( inf_inf @ assn @ A3 @ B2 ) @ C2 ) ) ).
% ent_conjE2
thf(fact_346_ent__conjE1,axiom,
! [A3: assn,C2: assn,B2: assn] :
( ( entails @ A3 @ C2 )
=> ( entails @ ( inf_inf @ assn @ A3 @ B2 ) @ C2 ) ) ).
% ent_conjE1
thf(fact_347_ent__conjI,axiom,
! [A3: assn,B2: assn,C2: assn] :
( ( entails @ A3 @ B2 )
=> ( ( entails @ A3 @ C2 )
=> ( entails @ A3 @ ( inf_inf @ assn @ B2 @ C2 ) ) ) ) ).
% ent_conjI
thf(fact_348_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A4: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A4 ) @ ( inf_inf @ A @ X @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_349_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A4: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A4 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_350_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A4: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A4 ) @ ( sup_sup @ A @ X @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_351_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A4: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A4 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_352_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F4: A > B,A6: A,B5: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F4 @ ( product_Pair @ A @ A @ A6 @ B5 ) ) ) ).
% pairself.cases
thf(fact_353_bex2I,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( P @ A4 @ B3 ) )
=> ? [A6: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ S )
& ( P @ A6 @ B5 ) ) ) ) ).
% bex2I
thf(fact_354_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_355_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_356_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_357_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_358_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% mult_minus_right
thf(fact_359_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B3 ) )
= ( times_times @ A @ A4 @ B3 ) ) ) ).
% minus_mult_minus
thf(fact_360_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% mult_minus_left
thf(fact_361_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_362_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_363_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_364_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_365_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_mult_commute
thf(fact_366_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ A4 )
= ( times_times @ A @ B3 @ B3 ) )
= ( ( A4 = B3 )
| ( A4
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% square_eq_iff
thf(fact_367_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_368_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_369_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A4: B,B3: C] :
( ( uncurry @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( F2 @ A4 @ B3 ) ) ).
% uncurry_apply
thf(fact_370_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 ) )
=> ~ ! [A6: A,B5: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B5 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B5 ) ) )
=> ~ ( 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 @ A6 @ B5 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_371_mod__pure,axiom,
! [B3: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( pure_assn @ B3 ) @ H2 )
= ( ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
= ( bot_bot @ ( set @ nat ) ) )
& B3 ) ) ).
% mod_pure
thf(fact_372_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_373_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H3: A,As3: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) )
=> ( ( Y
= ( ( As3
= ( bot_bot @ ( set @ B ) ) )
& X ) )
=> ~ ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_374_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H3: A,As3: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) ) )
=> ~ ( ( As3
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_375_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H3: A,As3: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As3 ) ) )
=> ( ( As3
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_376_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_377_mod__emp,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( one_one @ assn ) @ H2 )
= ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp
thf(fact_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,A4: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ A4 @ X )
= Zero )
=> ( ( ( Disj @ A4 @ X )
= One )
=> ( ( ( Conj @ A4 @ Y )
= Zero )
=> ( ( ( Disj @ A4 @ Y )
= One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
thf(fact_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A4: A,B3: B,P: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( P @ ( product_snd @ A @ B @ X ) )
=> ( P @ B3 ) ) ) ).
% sndE
thf(fact_401_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A4: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_402_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_403_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 )
=> ~ ! [A6: A,B5: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B5 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B5 ) ) ) ) ) ).
% pairself.elims
thf(fact_404_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: A,B3: A] :
( ( pairself @ A @ B @ F2 @ ( product_Pair @ A @ A @ A4 @ B3 ) )
= ( product_Pair @ B @ B @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) ) ).
% pairself.simps
thf(fact_405_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_406_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P5: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P5 ) )
= ( ? [A5: B] :
( P5
= ( product_Pair @ B @ A @ A5 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_407_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_408_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_409_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F: A > A > A,Z3: A] :
( ( abel_semigroup @ A @ F )
& ( comm_monoid_axioms @ A @ F @ Z3 ) ) ) ) ).
% comm_monoid_def
thf(fact_410_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_411_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_412_semilattice__neutr__def,axiom,
! [A: $tType] :
( ( semilattice_neutr @ A )
= ( ^ [F: A > A > A,Z3: A] :
( ( semilattice @ A @ F )
& ( comm_monoid @ A @ F @ Z3 ) ) ) ) ).
% semilattice_neutr_def
thf(fact_413_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_414_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A5: assn,B4: assn] : ( inf_inf @ assn @ A5 @ ( uminus_uminus @ assn @ B4 ) ) ) ) ).
% minus_assn_def
thf(fact_415_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)
thf(fact_416_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_417_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_418_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A8: A > B,B7: A > B,X2: A] : ( minus_minus @ B @ ( A8 @ X2 ) @ ( B7 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_419_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_420_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_421_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_422_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( ( A4
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_423_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ( times_times @ A @ C3 @ A4 )
= ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_left
thf(fact_424_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ( times_times @ A @ A4 @ C3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_right
thf(fact_425_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_426_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_0_right
thf(fact_427_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_428_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_zero
thf(fact_429_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_430_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_431_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_432_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_433_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A4 ) )
= ( ( zero_zero @ A )
= A4 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_434_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_435_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( A4
= ( uminus_uminus @ A @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_436_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ( uminus_uminus @ A @ A4 )
= A4 )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_437_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_438_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ ( inf_inf @ A @ B3 @ C3 ) )
= ( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_439_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_440_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A4 )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% sup.bounded_iff
thf(fact_441_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A4 @ B3 ) )
= ( minus_minus @ A @ B3 @ A4 ) ) ) ).
% minus_diff_eq
thf(fact_442_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_443_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_444_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_445_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_446_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_le_0_iff_le
thf(fact_447_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_448_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_449_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A4: A,C3: A] :
( ( ( times_times @ A @ A4 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_450_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B3: A] :
( ( C3
= ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_451_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A4: A] :
( ( ( times_times @ A @ C3 @ A4 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_452_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B3: A] :
( ( C3
= ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_453_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_454_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
= ( uminus_uminus @ A @ A4 ) ) ) ).
% diff_0
thf(fact_455_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_456_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A] :
( ( semilattice @ A @ F2 )
=> ( abel_semigroup @ A @ F2 ) ) ).
% semilattice.axioms(1)
thf(fact_457_semilattice_Oidem,axiom,
! [A: $tType,F2: A > A > A,A4: A] :
( ( semilattice @ A @ F2 )
=> ( ( F2 @ A4 @ A4 )
= A4 ) ) ).
% semilattice.idem
thf(fact_458_semilattice_Oleft__idem,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A] :
( ( semilattice @ A @ F2 )
=> ( ( F2 @ A4 @ ( F2 @ A4 @ B3 ) )
= ( F2 @ A4 @ B3 ) ) ) ).
% semilattice.left_idem
thf(fact_459_semilattice_Oright__idem,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A] :
( ( semilattice @ A @ F2 )
=> ( ( F2 @ ( F2 @ A4 @ B3 ) @ B3 )
= ( F2 @ A4 @ B3 ) ) ) ).
% semilattice.right_idem
thf(fact_460_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A8: A > B,B7: A > B,X2: A] : ( minus_minus @ B @ ( A8 @ X2 ) @ ( B7 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_461_abel__semigroup_Ocommute,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( F2 @ A4 @ B3 )
= ( F2 @ B3 @ A4 ) ) ) ).
% abel_semigroup.commute
thf(fact_462_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F2: A > A > A,B3: A,A4: A,C3: A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( F2 @ B3 @ ( F2 @ A4 @ C3 ) )
= ( F2 @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_463_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( A4 = B3 )
= ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_464_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_465_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ D3 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C3 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_466_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A4 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_467_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_468_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_469_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_470_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
= ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_471_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C3 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_472_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_473_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( ord_less_eq @ A @ A4 @ B3 ) )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( B3 != A4 ) ) ) ) ).
% nle_le
thf(fact_474_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_475_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_476_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_477_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_478_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_479_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% order.trans
thf(fact_480_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_481_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_482_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_483_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_484_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_485_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% antisym
thf(fact_486_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_487_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_488_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_489_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_490_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_491_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_492_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_493_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_494_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_495_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C3: B] :
( ( A4
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_496_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F2: A > B,C3: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F2 @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_497_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_498_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_499_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_500_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_501_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_502_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ A4 ) ) ) ).
% zero_le_square
thf(fact_503_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% split_mult_pos_le
thf(fact_504_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_505_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_506_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_left_mono
thf(fact_507_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_508_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_509_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_510_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_511_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_512_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_513_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_514_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B3 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_515_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_516_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_517_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_518_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_519_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_520_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] :
( ! [X2: A,Y2: A,Z3: A] :
( ( Conj2 @ X2 @ ( Disj2 @ Y2 @ Z3 ) )
= ( Disj2 @ ( Conj2 @ X2 @ Y2 ) @ ( Conj2 @ X2 @ Z3 ) ) )
& ! [X2: A,Y2: A,Z3: A] :
( ( Disj2 @ X2 @ ( Conj2 @ Y2 @ Z3 ) )
= ( Conj2 @ ( Disj2 @ X2 @ Y2 ) @ ( Disj2 @ X2 @ Z3 ) ) )
& ! [X2: A] :
( ( Conj2 @ X2 @ One2 )
= X2 )
& ! [X2: A] :
( ( Disj2 @ X2 @ Zero2 )
= X2 )
& ! [X2: A] :
( ( Conj2 @ X2 @ ( Compl2 @ X2 ) )
= Zero2 )
& ! [X2: A] :
( ( Disj2 @ X2 @ ( Compl2 @ X2 ) )
= One2 ) ) ) ) ).
% abstract_boolean_algebra_axioms_def
thf(fact_521_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,One: A,Zero: A,Compl: A > A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( Conj @ X3 @ ( Disj @ Y3 @ Z4 ) )
= ( Disj @ ( Conj @ X3 @ Y3 ) @ ( Conj @ X3 @ Z4 ) ) )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( Disj @ X3 @ ( Conj @ Y3 @ Z4 ) )
= ( Conj @ ( Disj @ X3 @ Y3 ) @ ( Disj @ X3 @ Z4 ) ) )
=> ( ! [X3: A] :
( ( Conj @ X3 @ One )
= X3 )
=> ( ! [X3: A] :
( ( Disj @ X3 @ Zero )
= X3 )
=> ( ! [X3: A] :
( ( Conj @ X3 @ ( Compl @ X3 ) )
= Zero )
=> ( ! [X3: A] :
( ( Disj @ X3 @ ( Compl @ X3 ) )
= One )
=> ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One ) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
thf(fact_522_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_523_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_524_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C3: A,A4: A] :
( ( ord_less_eq @ A @ C3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ A4 ) ) ) ) ).
% mult_left_le
thf(fact_525_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_526_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_527_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_528_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_529_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_530_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% left_diff_distrib
thf(fact_531_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ).
% right_diff_distrib
thf(fact_532_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( times_times @ A @ ( minus_minus @ A @ B3 @ C3 ) @ A4 )
= ( minus_minus @ A @ ( times_times @ A @ B3 @ A4 ) @ ( times_times @ A @ C3 @ A4 ) ) ) ) ).
% left_diff_distrib'
thf(fact_533_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ).
% right_diff_distrib'
thf(fact_534_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B3 ) @ A4 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ).
% minus_diff_commute
thf(fact_535_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_536_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
=> ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_537_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
= ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_538_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_539_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% subrelI
thf(fact_540_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_minus_iff
thf(fact_541_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% minus_le_iff
thf(fact_542_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_imp_neg_le
thf(fact_543_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_544_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_545_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_546_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_547_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_548_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_549_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_550_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_551_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_552_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_553_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A4 @ B3 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A4 )
=> ~ ( ord_less_eq @ A @ X @ B3 ) ) ) ) ).
% le_infE
thf(fact_554_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X @ A4 )
=> ( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% le_infI
thf(fact_555_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,C3: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ ( inf_inf @ A @ C3 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_556_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X ) ) ) ).
% le_infI1
thf(fact_557_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X ) ) ) ).
% le_infI2
thf(fact_558_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4
= ( inf_inf @ A @ A4 @ B3 ) ) ) ) ).
% inf.orderE
thf(fact_559_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( inf_inf @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% inf.orderI
thf(fact_560_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z4 )
=> ( ord_less_eq @ A @ X3 @ ( F2 @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_561_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( inf_inf @ A @ X2 @ Y2 )
= X2 ) ) ) ) ).
% le_iff_inf
thf(fact_562_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb1
thf(fact_563_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb2
thf(fact_564_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_565_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_566_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ ( inf_inf @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_567_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ord_less_eq @ A @ A4 @ ( inf_inf @ A @ B3 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_568_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_569_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_570_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ A4 ) ) ).
% inf.cobounded1
thf(fact_571_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ B3 ) ) ).
% inf.cobounded2
thf(fact_572_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_573_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_574_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_575_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_576_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_577_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_578_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A4 @ X )
=> ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% le_supE
thf(fact_579_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ X )
=> ( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X ) ) ) ) ).
% le_supI
thf(fact_580_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_581_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_582_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_583_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_584_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A4: A,D3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A4 )
=> ( ( ord_less_eq @ A @ D3 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D3 ) @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_585_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,C3: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ ( sup_sup @ A @ C3 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_586_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_587_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( sup_sup @ A @ X2 @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_588_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4
= ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_589_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% sup.orderI
thf(fact_590_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ ( F2 @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_591_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb1
thf(fact_592_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_593_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_594_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_595_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A4 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A4 )
=> ~ ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% sup.boundedE
thf(fact_596_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ A4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A4 ) ) ) ) ).
% sup.boundedI
thf(fact_597_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_598_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_599_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_600_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_601_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_602_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A4 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_603_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_604_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A4
!= ( zero_zero @ A ) )
& ( B3
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_605_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( ( A4
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_606_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ B3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_607_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A4 )
= ( times_times @ A @ C3 @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% mult_left_cancel
thf(fact_608_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( A4 = B3 ) ) ) ) ).
% mult_right_cancel
thf(fact_609_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_610_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_611_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( abel_semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.abel_semigroup_axioms
thf(fact_612_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( abel_semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.abel_semigroup_axioms
thf(fact_613_relH__subset,axiom,
! [Bs: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit,As: set @ nat] :
( ( relH @ Bs @ H2 @ H5 )
=> ( ( ord_less_eq @ ( set @ nat ) @ As @ Bs )
=> ( relH @ As @ H2 @ H5 ) ) ) ).
% relH_subset
thf(fact_614_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice @ A @ ( inf_inf @ A ) ) ) ).
% inf.semilattice_axioms
thf(fact_615_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice @ A @ ( sup_sup @ A ) ) ) ).
% sup.semilattice_axioms
thf(fact_616_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_617_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_618_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_619_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_620_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_621_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X2: A,Y2: A] : ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).
% diff_eq
thf(fact_622_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_623_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_624_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_625_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_626_disjoint__mono,axiom,
! [A: $tType,A4: set @ A,A7: set @ A,B3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A7 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_627_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_628_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_629_in__range__subset,axiom,
! [As: set @ nat,As2: set @ nat,H2: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% in_range_subset
thf(fact_630_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_631_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_632_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P5: A,Q4: A,R3: A] :
( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q4 @ R3 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q4 ) ) @ R3 ) ) ) ).
% sup_neg_inf
thf(fact_633_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_634_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_635_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_636_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X4: A,K3: A] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D4 ) ) )
| ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_637_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X4: A,K3: A] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D4 ) ) )
& ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_638_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_639_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_640_semilattice_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( semilattice_axioms @ A @ F2 )
=> ( semilattice @ A @ F2 ) ) ) ).
% semilattice.intro
thf(fact_641_semilattice__def,axiom,
! [A: $tType] :
( ( semilattice @ A )
= ( ^ [F: A > A > A] :
( ( abel_semigroup @ A @ F )
& ( semilattice_axioms @ A @ F ) ) ) ) ).
% semilattice_def
thf(fact_642_mod__star__trueE_H,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 )
= ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
=> ( ( ord_less_eq @ ( set @ nat ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
=> ~ ( rep_assn @ P @ H4 ) ) ) ) ).
% mod_star_trueE'
thf(fact_643_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_644_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_645_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A4: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_646_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_647_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_648_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_649_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_650_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_651_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% neg_less_iff_less
thf(fact_652_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_653_Diff__UNIV,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_654_Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_655_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_656_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_0_iff_less
thf(fact_657_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_658_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_pos
thf(fact_659_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_669_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_670_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_671_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F2: A > B,C3: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F2 @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_672_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C3: B] :
( ( A4
= ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_673_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_674_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order_less_asym'
thf(fact_675_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_676_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_677_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_678_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_679_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_680_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_681_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_682_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_683_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_684_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P6: A > $o] :
? [X5: A] : ( P6 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M: A] :
( ( ord_less @ A @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_685_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_686_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_687_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_688_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_689_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_690_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_691_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_692_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order.asym
thf(fact_693_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_694_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% dense
thf(fact_695_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_696_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_697_semilattice__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A] :
( ( F2 @ A6 @ A6 )
= A6 )
=> ( semilattice_axioms @ A @ F2 ) ) ).
% semilattice_axioms.intro
thf(fact_698_semilattice__axioms__def,axiom,
! [A: $tType] :
( ( semilattice_axioms @ A )
= ( ^ [F: A > A > A] :
! [A5: A] :
( ( F @ A5 @ A5 )
= A5 ) ) ) ).
% semilattice_axioms_def
thf(fact_699_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A,B5: A] :
( ( F2 @ A6 @ B5 )
= ( F2 @ B5 @ A6 ) )
=> ( abel_s757365448890700780axioms @ A @ F2 ) ) ).
% abel_semigroup_axioms.intro
thf(fact_700_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_701_disjoint__alt__simp3,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_702_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: A,B3: B,P: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( P @ ( product_fst @ A @ B @ X ) )
=> ( P @ A4 ) ) ) ).
% fstE
thf(fact_703_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A4: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_704_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_705_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_706_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A4: A,P5: product_prod @ A @ B] :
( ( A4
= ( product_fst @ A @ B @ P5 ) )
= ( ? [B4: B] :
( P5
= ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_707_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_708_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_709_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_710_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_711_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_712_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_713_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_714_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_715_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( A4 != B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% order_neq_le_trans
thf(fact_716_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% order_le_neq_trans
thf(fact_717_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_718_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_719_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_720_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% order_less_le
thf(fact_721_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% order_le_less
thf(fact_722_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_723_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_724_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_725_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_726_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_727_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_728_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_729_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_730_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_731_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_732_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_733_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_734_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_735_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_736_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_737_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_738_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_739_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_740_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_741_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_742_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( ord_less @ A @ A4 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% nless_le
thf(fact_743_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_744_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_745_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_746_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_747_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_748_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_749_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_750_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_751_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A4 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_752_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A4 @ B3 )
= ( ord_less @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_753_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ D3 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C3 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_754_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_755_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( A4
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A4 ) ) ) ).
% bot.not_eq_extremum
thf(fact_756_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_757_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_758_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_759_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% minus_less_iff
thf(fact_760_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% less_minus_iff
thf(fact_761_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% top.extremum_strict
thf(fact_762_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( A4
!= ( top_top @ A ) )
= ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_763_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_764_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ A4 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_765_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_766_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ ( inf_inf @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_767_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb4
thf(fact_768_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb3
thf(fact_769_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A4: A] :
( ( ord_less @ A @ B3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X ) ) ) ).
% less_infI2
thf(fact_770_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,X: A,B3: A] :
( ( ord_less @ A @ A4 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X ) ) ) ).
% less_infI1
thf(fact_771_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_772_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ A4 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_773_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_774_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A4 )
=> ~ ( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ C3 @ A4 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_775_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb4
thf(fact_776_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb3
thf(fact_777_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A4: A] :
( ( ord_less @ A @ X @ B3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_778_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A4: A,B3: A] :
( ( ord_less @ A @ X @ A4 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_779_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_780_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_781_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_782_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_783_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_784_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_785_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_786_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_787_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_788_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_789_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_790_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_791_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B3 @ A4 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_792_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_793_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_794_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B3 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_795_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_796_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_797_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_798_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_799_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( times_times @ A @ A4 @ A4 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_800_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_801_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_802_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_803_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_804_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_805_subset__minus__empty,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_806_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_807_disjoint__alt__simp2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
!= A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_808_disjoint__alt__simp1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_809_Int__Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_810_Diff__triv,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= A3 ) ) ).
% Diff_triv
thf(fact_811_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_812_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_813_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_814_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_815_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_816_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_817_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_818_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_819_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_820_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_821_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_822_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_823_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_824_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_825_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_826_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_827_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_828_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_829_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_830_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_831_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A] :
( ( semilattice @ A @ F2 )
=> ( semilattice_axioms @ A @ F2 ) ) ).
% semilattice.axioms(2)
thf(fact_832_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_833_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_834_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A4: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A4 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_835_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_836_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_837_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_838_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z4 )
=> ( ( ord_less @ A @ Z4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z4 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_839_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_840_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_841_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_842_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P5: A,Q: B > $o,Q4: B] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q4 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_843_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A4: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A4
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A4 ) @ ( product_snd @ A @ B @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_844_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_845_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_846_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A4 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_847_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_848_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_849_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( sgn_sgn @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_850_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A5: assn,B4: assn] :
( ( ord_less_eq @ assn @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% less_assn_def
thf(fact_851_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( ( ( F2 @ A4 @ B3 )
= Z2 )
= ( ( A4 = Z2 )
& ( B3 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_852_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( ( Z2
= ( F2 @ A4 @ B3 ) )
= ( ( A4 = Z2 )
& ( B3 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_853_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A4 @ Top ) ) ).
% ordering_top.extremum
thf(fact_854_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A4 ) ) ).
% ordering_top.extremum_strict
thf(fact_855_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
= ( A4 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_856_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A4 != Top )
= ( Less @ A4 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_857_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_858_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A,B3: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A4 ) @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% sgn_mult
thf(fact_859_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_860_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_861_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( sgn_sgn @ A @ A4 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% sgn_1_pos
thf(fact_862_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_863_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( sgn_sgn @ A @ A4 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_864_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A4: A,Y: A,U: A,V: A] :
( ( ord_less @ A @ X @ A4 )
=> ( ( ord_less @ A @ Y @ A4 )
=> ( ( 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 ) ) @ A4 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_865_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_866_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_867_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_868_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_869_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_870_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A4 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_871_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P7: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P7 ) @ ( product_fst @ A @ B @ P7 ) ) ) ) ).
% prod.swap_def
thf(fact_872_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_873_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C3 @ A4 ) )
= ( B3 = C3 ) ) ) ).
% add_right_cancel
thf(fact_874_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add_left_cancel
thf(fact_875_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_876_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_877_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% add_0
thf(fact_878_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_879_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_880_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( plus_plus @ A @ A4 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_881_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( plus_plus @ A @ B3 @ A4 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_882_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A,B3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= A4 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_883_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B3: A,A4: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= A4 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_884_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A4 @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_885_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% add.right_neutral
thf(fact_886_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_887_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_888_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% add_diff_cancel_right'
thf(fact_889_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_890_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ A4 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_891_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_892_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% diff_add_cancel
thf(fact_893_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% add_diff_cancel
thf(fact_894_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_add_distrib
thf(fact_895_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ ( plus_plus @ A @ A4 @ B3 ) )
= B3 ) ) ).
% minus_add_cancel
thf(fact_896_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ A4 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) )
= B3 ) ) ).
% add_minus_cancel
thf(fact_897_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% times_divide_eq_right
thf(fact_898_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( divide_divide @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ).
% divide_divide_eq_right
thf(fact_899_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 )
= ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_900_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( times_times @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A4 ) @ C3 ) ) ) ).
% times_divide_eq_left
thf(fact_901_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% div_by_1
thf(fact_902_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_903_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ A4 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_904_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_905_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel2
thf(fact_906_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel1
thf(fact_907_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_908_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B3 @ A4 ) @ B3 )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_909_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_910_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_911_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( plus_plus @ A @ B3 @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_912_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_913_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_914_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A4 ) @ B3 )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_915_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_916_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_917_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_918_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_919_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ A4 )
= B3 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_920_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_921_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_922_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_923_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_924_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_925_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ A4 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_926_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( ( divide_divide @ A @ A4 @ B3 )
= ( one_one @ A ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A4 = B3 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_927_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A4 @ B3 ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A4 = B3 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_928_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ A4 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_929_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( ( A4
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) )
& ( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_930_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ( divide_divide @ A @ B3 @ A4 )
= ( one_one @ A ) )
= ( ( A4
!= ( zero_zero @ A ) )
& ( A4 = B3 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_931_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B3 @ A4 ) )
= ( ( A4
!= ( zero_zero @ A ) )
& ( A4 = B3 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_932_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_933_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_934_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( minus_minus @ A @ B3 @ A4 ) ) ) ).
% uminus_add_conv_diff
thf(fact_935_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( plus_plus @ A @ A4 @ B3 ) ) ) ).
% diff_minus_eq_add
thf(fact_936_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_937_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( divide_divide @ A @ A4 @ ( sgn_sgn @ A @ B3 ) )
= ( times_times @ A @ A4 @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% divide_sgn
thf(fact_938_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_939_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_940_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_941_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_942_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_943_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_944_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_945_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_946_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_947_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_948_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_949_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B3 @ ( times_times @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A4 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_950_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_951_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_952_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_953_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C3 @ A4 ) )
=> ( B3 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_954_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C3 ) )
=> ( B3 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_955_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A4 @ C3 ) )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_956_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_957_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C3 @ A4 ) )
= ( B3 = C3 ) ) ) ).
% add.right_cancel
thf(fact_958_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add.left_cancel
thf(fact_959_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_960_add_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( plus_plus @ A @ A4 @ C3 ) @ B3 ) ) ) ).
% add.right_commute
thf(fact_961_add_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.right_assoc
thf(fact_962_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K: A,B3: A,A4: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_963_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A4: A,B3: A] :
( ( A3
= ( plus_plus @ A @ K @ A4 ) )
=> ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_964_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_965_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_966_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_967_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_968_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_969_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( 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_970_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= A4 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ Z2 ) @ B3 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_971_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= B3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_972_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B3 ) ) ) ).
% gt_half_sum
thf(fact_973_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ A4 @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_974_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_975_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_976_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_977_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_978_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ! [C4: A] :
( B3
!= ( plus_plus @ A @ A4 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_979_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_980_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_981_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_982_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_983_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_984_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% add.group_left_neutral
thf(fact_985_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% add.comm_neutral
thf(fact_986_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_987_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 )
= ( divide_divide @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_988_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z2: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_989_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_990_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_991_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_992_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_993_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_994_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_995_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_996_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_997_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_998_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,E3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ E3 ) @ C3 ) ) ) ).
% combine_common_factor
thf(fact_999_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% distrib_right
thf(fact_1000_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ).
% distrib_left
thf(fact_1001_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1002_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1003_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1004_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 )
= ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_1005_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ( plus_plus @ A @ C3 @ B3 )
= A4 )
=> ( C3
= ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_1006_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C3 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1007_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C3 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_1008_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( minus_minus @ A @ A4 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C3 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_1009_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_1010_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( A4
= ( minus_minus @ A @ C3 @ B3 ) )
= ( ( plus_plus @ A @ A4 @ B3 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_1011_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= C3 )
= ( A4
= ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_1012_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A4: A,B3: A] :
( ( A3
= ( plus_plus @ A @ K @ A4 ) )
=> ( ( minus_minus @ A @ A3 @ B3 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1013_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1014_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A4: A] :
( ( A3
= ( plus_plus @ A @ K @ A4 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1015_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_1016_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_1017_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_1018_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= B3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1019_add_Oac__operator__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( syntax_ac_operator @ A @ ( plus_plus @ A ) ) ) ).
% add.ac_operator_axioms
thf(fact_1020_add_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [X: A,Y: A,A4: A,B3: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A4 )
=> ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.safe_commute
thf(fact_1021_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R3: A,S2: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1022_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A4
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( times_times @ A @ A4 @ C3 )
= B3 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1023_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ C3 )
= A4 )
= ( B3
= ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1024_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C3 )
= B3 )
=> ( A4
= ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1025_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B3
= ( times_times @ A @ A4 @ C3 ) )
=> ( ( divide_divide @ A @ B3 @ C3 )
= A4 ) ) ) ) ).
% divide_eq_imp
thf(fact_1026_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1027_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ( divide_divide @ A @ B3 @ C3 )
= A4 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1028_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z2 ) )
= ( ( times_times @ A @ X @ Z2 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1029_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= ( one_one @ A ) )
= ( A4 = B3 ) ) ) ) ).
% right_inverse_eq
thf(fact_1030_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_1031_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_1032_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1033_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1034_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_1035_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% add_decreasing2
thf(fact_1036_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_1037_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% add_decreasing
thf(fact_1038_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1039_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1040_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_1041_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_1042_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1043_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ! [C4: A] :
( ( B3
= ( plus_plus @ A @ A4 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1044_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1045_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1046_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A4 )
= C3 )
= ( B3
= ( plus_plus @ A @ C3 @ A4 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1047_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ A4 ) )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1048_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1049_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A4 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1050_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1051_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B3 ) @ A4 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1052_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B3 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1053_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1054_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A4 ) ) ) ) ).
% le_add_diff
thf(fact_1055_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1056_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_1057_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 )
= ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_1058_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A] : ( ord_less @ A @ A4 @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_1059_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_1060_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ A4 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_1061_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 )
= ( ord_less @ A @ A4 @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_1062_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= B3 )
= ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1063_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( uminus_uminus @ A @ B3 ) )
= ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1064_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A4 )
= B3 ) ) ) ).
% add.inverse_unique
thf(fact_1065_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1066_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( B3
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% add_eq_0_iff
thf(fact_1067_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_1068_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,E3: A,C3: A,B3: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( C3
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E3 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1069_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,E3: A,C3: A,B3: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E3 ) @ C3 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1070_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_1071_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_1072_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,K: A,B3: A,A4: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1073_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.comm_monoid_axioms
thf(fact_1074_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_1075_sum__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.monoid_list_axioms
thf(fact_1076_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.comm_monoid_list_axioms
thf(fact_1077_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_list_set_axioms
thf(fact_1078_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_1079_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1080_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A4 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1081_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A4 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1082_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_1083_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_1084_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1085_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1086_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1087_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1088_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1089_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1090_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ B3 @ A4 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A4 @ B3 ) )
| ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1091_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1092_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_1093_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_1094_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( 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_1095_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= A4 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A4 @ Z2 ) @ B3 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1096_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1097_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1098_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1099_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1100_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1101_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1102_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_1103_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) ) )
= ( ( times_times @ A @ C3 @ B3 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1104_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) )
= C3 )
= ( ( uminus_uminus @ A @ A4 )
= ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1105_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) )
= A4 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B3 )
= ( times_times @ A @ A4 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1106_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4
= ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ C3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1107_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( ( divide_divide @ A @ A4 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A4
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1108_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_1109_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_1110_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_1111_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C3: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1112_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C3: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E3 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1113_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C3: A,B3: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1114_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C3: A,B3: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E3 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1115_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_1116_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1117_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A4 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1118_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_1119_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_1120_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1121_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1122_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1123_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1124_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A4 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1125_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1126_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1127_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B3 @ A4 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A4 @ B3 ) )
| ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1128_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1129_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( 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_1130_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( 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_1131_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A4 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1132_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1133_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1134_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1135_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1136_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1137_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_1138_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1139_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A4 ) @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1140_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A4: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A4 )
=> ( ( ord_less_eq @ A @ Y @ A4 )
=> ( ( 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 ) ) @ A4 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1141_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C3 ) @ A4 ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1142_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B3 ) @ A4 ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1143_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1144_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1145_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A4 ) ) ) ).
% div_minus1_right
thf(fact_1146_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1147_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% div_mult_mult2
thf(fact_1148_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% div_mult_mult1
thf(fact_1149_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% bits_div_by_1
thf(fact_1150_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1151_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A4 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1152_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_1153_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_1154_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_1155_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A4: A,B3: A,C3: A,D3: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A4 = B3 )
& ( C3 != D3 ) )
=> ( ( plus_plus @ A @ A4 @ ( times_times @ A @ R3 @ C3 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R3 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1156_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_1157_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ( A4 != B3 )
& ( C3 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A4 @ D3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1158_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1159_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1160_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A4: A,B3: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_1161_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_1162_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_1163_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B3: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1164_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A4: A,B3: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1165_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B3: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1166_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A4: A,B3: A,V: num] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( numeral_numeral @ A @ V ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1167_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1168_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1169_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1170_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A4 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1171_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1172_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) )
= A4 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1173_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( A4
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1174_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A4 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1175_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1176_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1177_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1178_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( A4
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1179_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A4 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1180_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1181_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1182_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_1183_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_1184_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_1185_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_1186_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_1187_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1188_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_le_numeral
thf(fact_1189_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1190_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1191_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1192_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C3 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1193_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1194_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_1195_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_less_numeral
thf(fact_1196_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_1197_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1198_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1199_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1200_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1201_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1202_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1203_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1204_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1205_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_1206_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_1207_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A4: A,B3: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_1208_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C3: A,A4: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_1209_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C3: A,A4: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_1210_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B3: A,A4: A] :
( ( nO_MATCH @ B @ A @ X @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_1211_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B3: A,A4: A] :
( ( nO_MATCH @ B @ A @ X @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A4 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_1212_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_1213_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_1214_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_1215_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_1216_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_1217_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_1218_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_1219_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_1220_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_1221_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_1222_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1223_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1224_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
= ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1225_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1226_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M2 ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1227_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_1228_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_1229_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_1230_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_1231_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_1232_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_1233_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_1234_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_1235_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_1236_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ( power_power @ A @ A4 @ M2 )
= ( power_power @ A @ A4 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_1237_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1238_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1239_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B3 @ A4 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1240_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ A4 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1241_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ A4 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_1242_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C3 ) @ A4 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self4
thf(fact_1243_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B3 ) @ A4 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self3
thf(fact_1244_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self2
thf(fact_1245_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self1
thf(fact_1246_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M2: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1247_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M2: num,N: num,B3: A] :
( ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ N ) ) @ B3 ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B3 ) ) ) ).
% power_add_numeral2
thf(fact_1248_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_1249_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_1250_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M2 ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1251_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_1252_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_1253_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ A4 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_1254_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_1255_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A4: 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 ) @ A4 ) )
= A4 ) ) ).
% left_minus_one_mult_self
thf(fact_1256_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ A4 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_1257_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( inverse_inverse @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_1258_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_1259_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1260_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_1261_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_1262_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ ( F2 @ A4 @ B3 ) )
= ( F2 @ ( Inverse @ B3 ) @ ( Inverse @ A4 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_1263_group_Ogroup__left__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ Z2 @ A4 )
= A4 ) ) ).
% group.group_left_neutral
thf(fact_1264_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_1265_group_Oinverse__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A4 ) )
= A4 ) ) ).
% group.inverse_inverse
thf(fact_1266_group_Oinverse__unique,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A4 @ B3 )
= Z2 )
=> ( ( Inverse @ A4 )
= B3 ) ) ) ).
% group.inverse_unique
thf(fact_1267_group_Oright__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ A4 @ ( Inverse @ A4 ) )
= Z2 ) ) ).
% group.right_inverse
thf(fact_1268_group_Oright__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,B3: A,A4: A,C3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ B3 @ A4 )
= ( F2 @ C3 @ A4 ) )
= ( B3 = C3 ) ) ) ).
% group.right_cancel
thf(fact_1269_group_Oleft__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ ( Inverse @ A4 ) @ A4 )
= Z2 ) ) ).
% group.left_inverse
thf(fact_1270_group_Oleft__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A,C3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A4 @ B3 )
= ( F2 @ A4 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% group.left_cancel
thf(fact_1271_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_1272_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A4 @ C3 ) @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% mod_mult_eq
thf(fact_1273_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C3: A,A7: A,B3: A,B6: A] :
( ( ( modulo_modulo @ A @ A4 @ C3 )
= ( modulo_modulo @ A @ A7 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C3 )
= ( modulo_modulo @ A @ B6 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A7 @ B6 ) @ C3 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_1274_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% mod_mult_mult2
thf(fact_1275_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( times_times @ A @ C3 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ).
% mult_mod_right
thf(fact_1276_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A4 @ C3 ) @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% mod_mult_left_eq
thf(fact_1277_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% mod_mult_right_eq
thf(fact_1278_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A4 @ N ) @ A4 )
= ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_commutes
thf(fact_1279_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A4 @ B3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1280_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_1281_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A4 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A4 )
= ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1282_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( power_power @ A @ A4 @ ( plus_plus @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_add
thf(fact_1283_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A4: A] :
( ( power_power @ A @ A4 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1284_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_1285_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A4 )
= B3 ) ) ) ).
% inverse_unique
thf(fact_1286_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_1287_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_1288_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_1289_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_1290_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ( modulo_modulo @ A @ A4 @ C3 )
= ( modulo_modulo @ A @ B3 @ C3 ) )
=> ~ ! [D2: A] :
( B3
!= ( plus_plus @ A @ A4 @ ( times_times @ A @ C3 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_1291_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1292_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% one_le_power
thf(fact_1293_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A4: A] :
( ( ord_less_eq @ nat @ N @ N3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ A4 @ N3 ) ) ) ) ) ).
% power_increasing
thf(fact_1294_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_1295_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1296_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1297_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A4: A] :
( ( ord_less @ nat @ N @ N3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ A4 @ N3 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1298_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P7: A,M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P7 @ ( power_power @ A @ P7 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1299_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_1300_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_one_over
thf(fact_1301_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_1302_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A4 )
= A4 ) ) ).
% mult_numeral_1
thf(fact_1303_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( numeral_numeral @ A @ one2 ) )
= A4 ) ) ).
% mult_numeral_1_right
thf(fact_1304_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_1305_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_1306_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% one_less_inverse
thf(fact_1307_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_1308_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ ( plus_plus @ A @ A4 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_1309_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% inverse_add
thf(fact_1310_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ A4 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_1311_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ ( minus_minus @ A @ B3 @ A4 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_1312_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A4 )
= ( divide_divide @ A @ ( one_one @ A ) @ A4 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_1313_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ C3 )
= ( plus_plus @ A @ A4 @ C3 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1314_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ C3 )
= ( plus_plus @ A @ A4 @ C3 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1315_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( A4
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ).
% mod_div_decomp
thf(fact_1316_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) )
= A4 ) ) ).
% div_mult_mod_eq
thf(fact_1317_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) )
= A4 ) ) ).
% mod_div_mult_eq
thf(fact_1318_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) )
= A4 ) ) ).
% mod_mult_div_eq
thf(fact_1319_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B3: A,A4: A] :
( ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) )
= A4 ) ) ).
% mult_div_mod_eq
thf(fact_1320_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 ) ) ) ) ).
% div_mult1_eq
thf(fact_1321_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1322_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1323_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1324_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1325_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A4 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1326_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A4: A] :
( ( ord_less_eq @ nat @ N @ N3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A4 @ N3 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1327_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1328_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A4: A] :
( ( ord_less @ nat @ N @ N3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N3 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1329_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1330_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1331_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,N: nat,M2: nat] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) )
= ( power_power @ A @ A4 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1332_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_minus
thf(fact_1333_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_1334_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_1335_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_1336_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ B3 @ A4 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_1337_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_1338_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% one_le_inverse
thf(fact_1339_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_1340_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_1341_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1342_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_1343_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_1344_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_1345_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_1346_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1347_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_1348_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_1349_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_1350_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X @ M2 ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1351_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_1352_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_1353_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_1354_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num] :
( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1355_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_1356_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_1357_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_1358_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_1359_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_1360_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1361_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1362_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_1363_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_1364_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_1365_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_1366_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_1367_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M: num,N2: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_1368_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_1369_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A4 @ A4 ) ) ) ).
% power2_eq_square
thf(fact_1370_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_1371_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_1372_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_1373_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ B3 ) ) ) ).
% left_add_twice
thf(fact_1374_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A4: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N ) )
= ( power_power @ A @ A4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_1375_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A4: A] :
( ( ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A4
= ( one_one @ A ) )
| ( A4
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_1376_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_1377_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_1378_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N2: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ) ).
% divmod_def
thf(fact_1379_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_1380_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_1381_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat,A4: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1382_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_1383_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_1384_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A4 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_1385_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A4 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_1386_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_1387_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_1388_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_1389_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_1390_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1391_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ A4 ) @ ( times_times @ A @ C3 @ A4 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1392_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C3 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1393_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1394_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1395_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ ( times_times @ A @ C3 @ A4 ) @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1396_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ ( times_times @ A @ C3 @ A4 ) ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1397_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1398_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% dvd_div_mult_self
thf(fact_1399_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ A4 ) )
= B3 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1400_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= A4 ) ) ) ).
% unit_div_1_div_1
thf(fact_1401_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1402_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1403_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_1404_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_1405_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B3 @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= ( divide_divide @ A @ B3 @ A4 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1406_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% unit_div_mult_self
thf(fact_1407_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A4 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_mult_iff
thf(fact_1408_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1409_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_1410_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_1411_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1412_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1413_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1414_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A4 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1415_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_1416_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1417_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A] : ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ A4 ) ) ) ).
% dvd_triv_right
thf(fact_1418_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
=> ( dvd_dvd @ A @ B3 @ C3 ) ) ) ).
% dvd_mult_right
thf(fact_1419_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1420_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A] : ( dvd_dvd @ A @ A4 @ ( times_times @ A @ A4 @ B3 ) ) ) ).
% dvd_triv_left
thf(fact_1421_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
=> ( dvd_dvd @ A @ A4 @ C3 ) ) ) ).
% dvd_mult_left
thf(fact_1422_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% dvd_mult2
thf(fact_1423_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C3 )
=> ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% dvd_mult
thf(fact_1424_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_1425_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A4: A,B3: A,K: A] :
( ( A4
= ( times_times @ A @ B3 @ K ) )
=> ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% dvdI
thf(fact_1426_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ~ ! [K2: A] :
( A4
!= ( times_times @ A @ B3 @ K2 ) ) ) ) ).
% dvdE
thf(fact_1427_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A4 ) ) ).
% one_dvd
thf(fact_1428_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% unit_imp_dvd
thf(fact_1429_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1430_not__is__unit__0,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% not_is_unit_0
thf(fact_1431_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1432_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1433_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A4 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1434_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ A4 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1435_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1436_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A4 @ B3 )
= ( times_times @ A @ A4 @ C3 ) )
= ( B3 = C3 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1437_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B3 @ A4 )
= ( times_times @ A @ C3 @ A4 ) )
= ( B3 = C3 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1438_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A4 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A4 ) @ C3 ) ) ) ) ).
% dvd_div_mult
thf(fact_1439_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ) ).
% div_mult_swap
thf(fact_1440_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1441_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ C3 ) @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1442_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 )
=> ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1443_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,D3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( dvd_dvd @ A @ D3 @ C3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( divide_divide @ A @ C3 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1444_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ C3 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ C3 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1445_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A4 @ C3 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1446_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= ( divide_divide @ A @ C3 @ A4 ) )
= ( B3 = C3 ) ) ) ) ).
% unit_div_cancel
thf(fact_1447_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1448_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ~ ( ( A4
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B3
!= ( times_times @ A @ A4 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1449_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X4: A,K3: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X4 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1450_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X4: A,K3: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1451_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= C3 )
= ( B3
= ( times_times @ A @ C3 @ A4 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1452_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1453_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1454_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C3: A,B3: A,D3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= ( divide_divide @ A @ D3 @ C3 ) )
= ( ( times_times @ A @ B3 @ C3 )
= ( times_times @ A @ A4 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1455_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1456_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= C3 )
= ( A4
= ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1457_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( A4
= ( divide_divide @ A @ C3 @ B3 ) )
= ( ( times_times @ A @ A4 @ B3 )
= C3 ) ) ) ) ).
% unit_eq_div2
thf(fact_1458_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1459_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 ) ) ) ) ).
% unit_div_commute
thf(fact_1460_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1461_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C3 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1462_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_1463_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A4 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1464_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ A4 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1465_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1466_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ~ ( ( A4
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A4 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A4 )
=> ( ( ( times_times @ A @ A4 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A4 )
!= ( times_times @ A @ C3 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1467_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ~ ! [B5: A] :
( A4
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% evenE
thf(fact_1468_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1469_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1470_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_1471_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A4 ) ) ) ).
% even_two_times_div_two
thf(fact_1472_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) )
= ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_1473_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ~ ! [B5: A] :
( A4
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1474_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_1475_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_1476_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_1477_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: 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 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_1478_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: 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 ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_1479_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1480_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_1481_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_1482_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_1483_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_1484_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_1485_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( 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 ) ) @ M2 )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_1486_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M: nat,N2: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_1487_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1488_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_1489_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_1490_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_1491_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_1492_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_1493_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_1494_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_1495_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_1496_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A4 ) @ ( sgn_sgn @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( A4
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_1497_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_1498_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_1499_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A4 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_1500_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_1501_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_1502_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_1503_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_1504_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_1505_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_1506_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_1507_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_1508_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_1509_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_1510_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_1511_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P5
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1512_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ( P5
=> ( P @ ( one_one @ A ) ) )
& ( ~ P5
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1513_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P7: $o] : ( if @ A @ P7 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_1514_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N4: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N4 ) ) )
=> ( ! [N4: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N4 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) )
=> ( ! [M3: num,N4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) )
=> ~ ! [M3: num,N4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_1515_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_Suc
thf(fact_1516_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ N ) @ A4 ) ) ) ).
% power_Suc2
thf(fact_1517_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_1518_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1519_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_1520_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_1521_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A4 @ ( suc @ N ) ) @ A4 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1522_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A4 @ A4 ) @ A4 ) ) ) ).
% power3_eq_cube
thf(fact_1523_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1524_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_1525_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_1526_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A4: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B5: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% bits_induct
thf(fact_1527_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ A4 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A4 @ ( power_power @ A @ ( power_power @ A @ A4 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_1528_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M2 @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% exp_mod_exp
thf(fact_1529_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_1530_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_1531_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,N: nat] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A4 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A4 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_1532_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1533_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1534_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A4 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1535_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1536_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_1537_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_1538_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_1539_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A4 )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ) ).
% one_xor_eq
thf(fact_1540_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A4 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ) ).
% xor_one_eq
thf(fact_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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_1556_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_1557_or_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semigroup @ A @ ( bit_se1065995026697491101ons_or @ A ) ) ) ).
% or.semigroup_axioms
thf(fact_1558_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_1559_xor_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semigroup @ A @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% xor.semigroup_axioms
thf(fact_1560_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.comm_monoid_axioms
thf(fact_1561_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_1562_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.semilattice_neutr_axioms
thf(fact_1563_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.comm_monoid_axioms
thf(fact_1564_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_1565_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_1566_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_1567_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se1065995026697491101ons_or @ A @ A4 @ ( one_one @ A ) )
= ( plus_plus @ A @ A4 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ) ).
% or_one_eq
thf(fact_1568_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A4 )
= ( plus_plus @ A @ A4 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ) ).
% one_or_eq
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_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_1571_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_1572_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_1573_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,B3: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A4 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_1574_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% diff_numeral_special(8)
thf(fact_1575_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_1576_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_1577_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_1578_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_1579_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A] :
( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A4 ) ) ).
% and.right_neutral
thf(fact_1580_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A4 )
= A4 ) ) ).
% and.left_neutral
thf(fact_1581_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_1582_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_1583_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_1584_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_1585_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_1586_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_1587_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_1588_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_1589_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_1590_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_1591_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_1592_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_1593_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_1594_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_1595_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_1596_and_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semigroup @ A @ ( bit_se5824344872417868541ns_and @ A ) ) ) ).
% and.semigroup_axioms
thf(fact_1597_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_1598_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,B3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A4 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A4
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A4 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A4 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A4 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A4 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_1606_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A4 )
= ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_1607_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_1608_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_1609_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_1610_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_1611_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_1612_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_1613_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P5: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ P5 @ ( times_times @ A @ A4 @ B3 ) )
=> ~ ! [X3: A,Y3: A] :
( ( P5
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( dvd_dvd @ A @ X3 @ A4 )
=> ~ ( dvd_dvd @ A @ Y3 @ B3 ) ) ) ) ) ).
% dvd_productE
thf(fact_1614_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
=> ? [B8: A,C6: A] :
( ( A4
= ( times_times @ A @ B8 @ C6 ) )
& ( dvd_dvd @ A @ B8 @ B3 )
& ( dvd_dvd @ A @ C6 @ C3 ) ) ) ) ).
% division_decomp
thf(fact_1615_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M2: 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 ) ) @ M2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_1616_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_1617_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F: A > A > A,Z3: A,Inverse2: A > A] :
( ( semigroup @ A @ F )
& ( group_axioms @ A @ F @ Z3 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_1618_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A4 @ B3 ) @ B3 )
= ( ord_min @ A @ A4 @ B3 ) ) ) ).
% min.right_idem
thf(fact_1619_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_min @ A @ A4 @ ( ord_min @ A @ A4 @ B3 ) )
= ( ord_min @ A @ A4 @ B3 ) ) ) ).
% min.left_idem
thf(fact_1620_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A] :
( ( ord_min @ A @ A4 @ A4 )
= A4 ) ) ).
% min.idem
thf(fact_1621_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) )
= ( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_1622_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb2
thf(fact_1623_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb1
thf(fact_1624_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_1625_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb4
thf(fact_1626_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb3
thf(fact_1627_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_1628_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_1629_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_1630_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_1631_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_1632_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_1633_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_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_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_1642_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_1643_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_1644_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_1645_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F: A > A > A,Z3: A,Inverse2: A > A] :
( ! [A5: A] :
( ( F @ Z3 @ A5 )
= A5 )
& ! [A5: A] :
( ( F @ ( Inverse2 @ A5 ) @ A5 )
= Z3 ) ) ) ) ).
% group_axioms_def
thf(fact_1646_group__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ! [A6: A] :
( ( F2 @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F2 @ ( Inverse @ A6 ) @ A6 )
= Z2 )
=> ( group_axioms @ A @ F2 @ Z2 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_1647_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_min @ A @ B3 @ ( ord_min @ A @ A4 @ C3 ) )
= ( ord_min @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ).
% min.left_commute
thf(fact_1648_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( ord_min @ A @ B4 @ A5 ) ) ) ) ).
% min.commute
thf(fact_1649_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A4 @ B3 ) @ C3 )
= ( ord_min @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ).
% min.assoc
thf(fact_1650_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_1651_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_1652_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_1653_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_1654_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_1655_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_1656_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_1657_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_1658_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ B3 ) ) ).
% min.cobounded2
thf(fact_1659_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ A4 ) ) ).
% min.cobounded1
thf(fact_1660_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_1661_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ord_less_eq @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_1662_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_1663_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( ord_min @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% min.orderI
thf(fact_1664_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4
= ( ord_min @ A @ A4 @ B3 ) ) ) ) ).
% min.orderE
thf(fact_1665_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,C3: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ ( ord_min @ A @ C3 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_1666_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% min.strict_coboundedI2
thf(fact_1667_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( ord_less @ A @ A4 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ C3 ) ) ) ).
% min.strict_coboundedI1
thf(fact_1668_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_1669_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_less @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ A4 @ C3 ) ) ) ) ).
% min.strict_boundedE
thf(fact_1670_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_1671_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_1672_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_1673_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_1674_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_1675_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.abel_semigroup_axioms
thf(fact_1676_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_min @ A ) ) ) ).
% min.semilattice_axioms
thf(fact_1677_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.semigroup_axioms
thf(fact_1678_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_1679_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_1680_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_1681_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_1682_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_1683_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_1684_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_1685_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_1686_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_1687_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_1688_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_1689_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_1690_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_1691_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_1692_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_1693_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 ) ) ) )
=> ( ! [F4: nat > A > A,A6: nat,B5: 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 ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B5 @ A6 )
=> ( P @ F4 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F4 @ A6 @ Acc ) ) )
=> ( P @ F4 @ A6 @ B5 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_1694_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A4 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_1695_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) )
=> ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_1696_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_1697_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A4 ) )
= ( abs_abs @ A @ A4 ) ) ) ).
% abs_idempotent
thf(fact_1698_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_1699_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ( abs_abs @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_1700_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1701_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_add_abs
thf(fact_1702_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ A4 ) )
= ( times_times @ A @ A4 @ A4 ) ) ) ).
% abs_mult_self_eq
thf(fact_1703_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_1704_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A4 ) )
= ( abs_abs @ A @ A4 ) ) ) ).
% abs_minus_cancel
thf(fact_1705_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( abs_abs @ A @ A4 )
= A4 ) ) ) ).
% abs_of_nonneg
thf(fact_1706_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ A4 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% abs_le_self_iff
thf(fact_1707_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_1708_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A4 ) )
= ( A4
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_1709_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_1710_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A4: A] :
( ( gbinomial @ A @ A4 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1711_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_1712_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A4 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% abs_of_nonpos
thf(fact_1713_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_1714_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_1715_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_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% abs_le_D1
thf(fact_1727_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( abs_abs @ A @ A4 ) ) ) ).
% abs_ge_self
thf(fact_1728_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A,B3: A] :
( ( abs_abs @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_mult
thf(fact_1729_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_1730_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ).
% abs_minus_commute
thf(fact_1731_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A4 ) ) ) ).
% abs_ge_zero
thf(fact_1732_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A4 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_1733_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( abs_abs @ A @ A4 )
= A4 ) ) ) ).
% abs_of_pos
thf(fact_1734_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A4 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1735_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_1736_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_1737_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1738_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C3: A,B3: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A4 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B3 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) @ ( times_times @ A @ C3 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1739_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 ) ) ) ) ).
% abs_leI
thf(fact_1740_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ).
% abs_le_D2
thf(fact_1741_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 )
= ( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ) ).
% abs_le_iff
thf(fact_1742_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( abs_abs @ A @ A4 ) ) ) ).
% abs_ge_minus_self
thf(fact_1743_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_1744_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A4 @ K ) @ ( gbinomial @ A @ A4 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_1745_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_1746_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A4 ) @ ( abs_abs @ A @ A4 ) )
= A4 ) ) ).
% sgn_mult_abs
thf(fact_1747_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( sgn_sgn @ A @ A4 ) )
= A4 ) ) ).
% abs_mult_sgn
thf(fact_1748_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_1749_euclidean__size__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ A4 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% euclidean_size_unit
thf(fact_1750_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% euclidean_size_mult
thf(fact_1751_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( gbinomial @ A @ A4 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_1752_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_1753_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A4: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
| ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
| ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1754_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_1755_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A4 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_1756_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1757_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ C3 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1758_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A4 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% abs_of_neg
thf(fact_1759_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( A4
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( A4
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_1760_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A4 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A4
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_1761_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% size_mult_mono
thf(fact_1762_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B3 @ A4 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_1763_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_1764_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_1765_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F4: nat > A > A,A6: nat,B5: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1766_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_1767_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_1768_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A4: nat,B3: 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 ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B3 @ A4 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A4 @ B3 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B3 @ A4 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A4 @ B3 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A4 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_1769_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,A4: A] :
( ( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( A4
!= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) ) ) )
=> ( ( ( B3
!= ( zero_zero @ A ) )
=> ! [Q6: A,R4: A] :
( ( ( euclid7384307370059645450egment @ A @ R4 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R4 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( R4
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= Q6 )
=> ( ( ( modulo_modulo @ A @ A4 @ B3 )
= R4 )
=> ( A4
!= ( plus_plus @ A @ ( times_times @ A @ Q6 @ B3 ) @ R4 ) ) ) ) ) ) ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_1770_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A4 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1771_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R3: A,Q4: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R3 )
= A4 )
=> ( ( modulo_modulo @ A @ A4 @ B3 )
= R3 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_1772_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R3: A,Q4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R3 ) @ B3 )
= Q4 ) ) ) ) ) ).
% div_bounded
thf(fact_1773_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_1774_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_1775_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_1776_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A4 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A4 ) ) )
= A4 ) ) ).
% division_segment_euclidean_size
thf(fact_1777_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( divide_divide @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_1778_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_1779_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A4 ) @ ( euclid7384307370059645450egment @ A @ B3 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_1780_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A4 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_1781_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A4 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1782_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A4 @ K ) )
= ( times_times @ A @ A4 @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1783_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( times_times @ A @ A4 @ ( gbinomial @ A @ A4 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A4 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A4 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1784_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A4 @ K ) @ A4 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A4 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A4 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1785_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A4: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A4 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_1786_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A4: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A4 @ ( suc @ K ) ) )
= ( times_times @ A @ A4 @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_1787_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M2: nat,A4: A] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A4 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A4 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M2 @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_1788_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A4 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_1789_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A4 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_1790_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_1791_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_1792_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A4 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_1793_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R3: A,Q4: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R3 )
= A4 )
=> ( ( divide_divide @ A @ A4 @ B3 )
= Q4 ) ) ) ) ) ) ).
% div_eqI
thf(fact_1794_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% of_nat_Suc
thf(fact_1795_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_1796_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_1797_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_1798_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M2 @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_1799_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N4: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_1800_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N4: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1801_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_1802_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_1803_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_1804_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( 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 @ B3 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_1805_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_1806_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_1807_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_1808_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_1809_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_1810_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_1811_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_1812_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_1813_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_1814_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_1815_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_1816_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] :
! [X2: A,Y2: A] :
( ( Xor2 @ X2 @ Y2 )
= ( Disj2 @ ( Conj2 @ X2 @ ( Compl2 @ Y2 ) ) @ ( Conj2 @ ( Compl2 @ X2 ) @ Y2 ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
thf(fact_1817_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] :
( ! [X3: A,Y3: A] :
( ( Xor @ X3 @ Y3 )
= ( Disj @ ( Conj @ X3 @ ( Compl @ Y3 ) ) @ ( Conj @ ( Compl @ X3 ) @ Y3 ) ) )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
thf(fact_1818_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_1819_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_1820_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_1821_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_1822_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_1823_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_1824_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_1825_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_1826_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ A4 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_1827_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_1828_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A4 @ N ) @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_1829_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_1830_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_1831_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_1832_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_1833_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_1834_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A4: A,B3: A] :
( ( ( finite_card @ A @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A4 != B3 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_singletonI,axiom,
! [A: $tType,A4: A] : ( member @ A @ A4 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_1842_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_1843_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1844_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A4: A,A3: set @ A] :
( ( ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ A4 @ A3 ) )
= ( ( A4 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1845_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A4: A,A3: set @ A,B3: A] :
( ( ( insert2 @ A @ A4 @ A3 )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A4 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1846_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set @ A,B3: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ B2 ) ) )
= ( ~ ( member @ A @ B3 @ A3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1847_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A4: A,A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert2 @ A @ A4 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A4 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_1848_insert__disjoint_I2_J,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ A3 ) @ B2 ) )
= ( ~ ( member @ A @ A4 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1849_insert__disjoint_I1_J,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ A3 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A4 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_1850_insert__Diff__single,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( insert2 @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert2 @ A @ A4 @ A3 ) ) ).
% insert_Diff_single
thf(fact_1851_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_1852_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_1853_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_1854_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_1855_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_1856_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_1857_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_1858_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_1859_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_1860_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_1861_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_1862_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_1863_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_1864_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_1865_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_1866_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_1867_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_1868_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_1869_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M: num,N2: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_1870_singleton__inject,axiom,
! [A: $tType,A4: A,B3: A] :
( ( ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A4 = B3 ) ) ).
% singleton_inject
thf(fact_1871_insert__not__empty,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( insert2 @ A @ A4 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_1872_doubleton__eq__iff,axiom,
! [A: $tType,A4: A,B3: A,C3: A,D3: A] :
( ( ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C3 @ ( insert2 @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A4 = C3 )
& ( B3 = D3 ) )
| ( ( A4 = D3 )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1873_singleton__iff,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A4 ) ) ).
% singleton_iff
thf(fact_1874_singletonD,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A4 ) ) ).
% singletonD
thf(fact_1875_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_1876_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_1877_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_1878_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_1879_subset__singleton__iff,axiom,
! [A: $tType,X6: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
| ( X6
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_1880_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_1881_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1882_singleton__Un__iff,axiom,
! [A: $tType,X: A,A3: set @ A,B2: set @ A] :
( ( ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1883_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A4: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_1884_insert__Diff,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( insert2 @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1885_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A4: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1886_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1887_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X6: set @ A] :
( ~ ( member @ A @ X @ X6 )
=> ( ( minus_minus @ ( set @ A ) @ X6 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X6 ) ) ).
% set_minus_singleton_eq
thf(fact_1888_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_1889_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A4 ) @ ( archim6421214686448440834_floor @ A @ B3 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_1890_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A4 @ B3 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A4 ) @ ( archimedean_ceiling @ A @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1891_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1892_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1893_card__1__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A3
!= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_1894_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_1895_Compl__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_1896_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B4: A,B7: set @ A] :
( ( A3
= ( insert2 @ A @ B4 @ B7 ) )
& ~ ( member @ A @ B4 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1897_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
=> ? [B5: A,B9: set @ A] :
( ( A3
= ( insert2 @ A @ B5 @ B9 ) )
& ~ ( member @ A @ B5 @ B9 )
& ( ( finite_card @ A @ B9 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_1898_card__1__singleton__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A3
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1899_card__Diff1__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ).
% card_Diff1_le
thf(fact_1900_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ B2 )
=> ( ord_less @ ( set @ A ) @ A3 @ B2 ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1901_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y2: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y2
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X2 @ Y2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X2 @ Y2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X2 @ Y2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_1902_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_1903_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y2: A] :
( ( S
= ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y2 ) ) ) ) ).
% card_2_iff
thf(fact_1904_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y2: A,Z3: A] :
( ( S
= ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y2 )
& ( Y2 != Z3 )
& ( X2 != Z3 ) ) ) ) ).
% card_3_iff
thf(fact_1905_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1906_card__Diff__singleton,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_1907_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_1908_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A4 @ ( one_one @ int ) ) @ B3 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( 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_1909_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_1910_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_1911_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( 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_1912_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_1913_divmod__BitM__2__eq,axiom,
! [M2: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_1914_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_1915_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_1916_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_1917_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_1918_unique__quotient,axiom,
! [A4: int,B3: int,Q4: int,R3: int,Q7: int,R5: int] :
( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q7 @ R5 ) )
=> ( Q4 = Q7 ) ) ) ).
% unique_quotient
thf(fact_1919_unique__remainder,axiom,
! [A4: int,B3: int,Q4: int,R3: int,Q7: int,R5: int] :
( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q7 @ R5 ) )
=> ( R3 = R5 ) ) ) ).
% unique_remainder
thf(fact_1920_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_1921_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
( A8
= ( insert2 @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1922_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1923_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_1924_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_1925_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,Q8: int] :
( ( A12 = K4 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q8 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K4
= ( times_times @ int @ Q8 @ L2 ) ) )
| ? [R2: int,L2: int,K4: int,Q8: int] :
( ( A12 = K4 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q8 @ R2 ) )
& ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L2 ) )
& ( K4
= ( plus_plus @ int @ ( times_times @ int @ Q8 @ L2 ) @ R2 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_1926_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q6: int] :
( ( A32
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q6 @ A22 ) ) ) )
=> ~ ! [R4: int,Q6: int] :
( ( A32
= ( product_Pair @ int @ int @ Q6 @ R4 ) )
=> ( ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q6 @ A22 ) @ R4 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_1927_zminus1__lemma,axiom,
! [A4: int,B3: int,Q4: int,R3: int] :
( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( B3
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A4 ) @ B3
@ ( 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 @ B3 @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_1928_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_1929_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
? [X2: A] :
( A8
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_1930_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X3: A] :
( A3
!= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_1931_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_1932_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_1933_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_1934_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_1935_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_1936_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_1937_normalize__negative,axiom,
! [Q4: int,P5: int] :
( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_1938_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_1939_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_1940_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_1941_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_1942_normalize__denom__zero,axiom,
! [P5: int] :
( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I2: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I2 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% floor_split
thf(fact_1954_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A4: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A4 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A4 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1955_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_1956_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_1957_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_1958_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A4: int] :
( ( ( archimedean_ceiling @ A @ X )
= A4 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A4 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A4 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1959_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I2: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% ceiling_split
thf(fact_1960_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_1961_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_1962_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_1963_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_1964_normalize__denom__pos,axiom,
! [R3: product_prod @ int @ int,P5: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_1965_normalize__crossproduct,axiom,
! [Q4: int,S2: int,P5: int,R3: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S2
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R3 @ S2 ) ) )
=> ( ( times_times @ int @ P5 @ S2 )
= ( times_times @ int @ R3 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_1966_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_1967_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_1968_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: 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 @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ) ).
% floor_divide_lower
thf(fact_1969_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1970_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_1971_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: 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 @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P5 ) ) ) ).
% ceiling_divide_lower
thf(fact_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1979_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A4: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A4 )
=> ( ( member @ B @ A4 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A4 @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A4 ) @ ( archimedean_ceiling @ B @ B3 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_1980_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A4: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A4 )
=> ( ( member @ B @ A4 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A4 ) @ ( archim6421214686448440834_floor @ B @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A4 @ B3 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_1981_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A4: A] :
( ( ( archimedean_frac @ A @ X )
= A4 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A4 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ A4 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_1982_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( gcd_Gcd @ A @ A3 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_1983_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A8: set @ A] : ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_1984_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_1985_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_1986_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_1987_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_1988_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_1989_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_1990_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_1991_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,B3: A] :
( ( member @ A @ A4 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A4 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_1992_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_1993_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_1994_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A3 )
=> ( ( gcd_Gcd @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_1995_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( gcd_Gcd @ A @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_1996_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A4: A] :
( ( member @ A @ A4 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ A4 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_1997_quotient__of__denom__pos,axiom,
! [R3: rat,P5: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_1998_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( member @ A @ A4 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_1999_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_2000_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_2001_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_2002_rat__sgn__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_2003_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_2004_quotient__of__int,axiom,
! [A4: int] :
( ( quotient_of @ ( of_int @ A4 ) )
= ( product_Pair @ int @ int @ A4 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_2005_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_2006_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_2007_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_2008_Frct__code__post_I8_J,axiom,
! [A4: int,B3: int] :
( ( frct @ ( product_Pair @ int @ int @ A4 @ ( uminus_uminus @ int @ B3 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A4 @ B3 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_2009_Frct__code__post_I7_J,axiom,
! [A4: int,B3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A4 ) @ B3 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A4 @ B3 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_2010_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_2011_Frct__code__post_I1_J,axiom,
! [A4: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A4 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_2012_Frct__code__post_I2_J,axiom,
! [A4: int] :
( ( frct @ ( product_Pair @ int @ int @ A4 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_2013_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_2014_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_2015_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_2016_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_2017_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I2: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I2 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I2 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2018_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_2019_card__Un__disjoint,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( finite_finite @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_2020_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_2021_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_2022_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_2023_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_2024_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
( ( set_or1337092689740270186AtMost @ A @ A4 @ A4 )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_2025_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A4 = B3 )
& ( B3 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_2026_of__rat__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
= ( one_one @ A ) ) ) ).
% of_rat_1
thf(fact_2027_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: rat] :
( ( ( field_char_0_of_rat @ A @ A4 )
= ( one_one @ A ) )
= ( A4
= ( one_one @ rat ) ) ) ) ).
% of_rat_eq_1_iff
thf(fact_2028_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: rat] :
( ( ( one_one @ A )
= ( field_char_0_of_rat @ A @ A4 ) )
= ( ( one_one @ rat )
= A4 ) ) ) ).
% one_eq_of_rat_iff
thf(fact_2029_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_2030_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_2031_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_2032_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_2033_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_2034_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: rat,B3: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A4 @ B3 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A4 ) @ ( field_char_0_of_rat @ A @ B3 ) ) ) ) ).
% of_rat_mult
thf(fact_2035_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_2036_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_2037_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2038_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_2039_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_2040_finite_Ocases,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A9: set @ A] :
( ? [A6: A] :
( A4
= ( insert2 @ A @ A6 @ A9 ) )
=> ~ ( finite_finite @ A @ A9 ) ) ) ) ).
% finite.cases
thf(fact_2041_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,B4: A] :
( ( A5
= ( insert2 @ A @ B4 @ A8 ) )
& ( finite_finite @ A @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_2042_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_2043_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X3 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_2044_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ! [A9: set @ A] :
( ~ ( finite_finite @ A @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_2045_Gcd__remove0__nat,axiom,
! [M4: set @ nat] :
( ( finite_finite @ 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_2046_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_2047_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite @ A @ F6 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_2048_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite @ A @ F6 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_2049_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: A,A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ( member @ A @ A6 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_2050_infinite__coinduct,axiom,
! [A: $tType,X6: ( set @ A ) > $o,A3: set @ A] :
( ( X6 @ A3 )
=> ( ! [A9: set @ A] :
( ( X6 @ A9 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A9 )
& ( ( X6 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_2051_infinite__remove,axiom,
! [A: $tType,S: set @ A,A4: A] :
( ~ ( finite_finite @ A @ S )
=> ~ ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_2052_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_2053_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_2054_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B2: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite @ A @ B2 )
=> ( P @ B2 ) )
=> ( ! [A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_2055_finite__remove__induct,axiom,
! [A: $tType,B2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ B2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_2056_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_2057_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X: A] :
( ( finite_finite @ 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_2058_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T3: set @ A] :
( ( ord_less @ ( set @ A ) @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ S @ T3 ) )
& ( P @ ( insert2 @ A @ X4 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_2059_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_2060_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_2061_card__Suc__Diff1,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_2062_card_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_2063_card_Oremove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_2064_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) )
= ( ( finite_finite @ A @ A3 )
& ( member @ A @ X @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_2065_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X: A,Y: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_2066_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( ord_less @ A @ X4 @ B5 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert2 @ A @ B5 @ A9 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_2067_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( ord_less @ A @ B5 @ X4 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert2 @ A @ B5 @ A9 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_2068_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S4: set @ B] :
( ( finite_finite @ B @ S4 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S4 )
=> ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert2 @ B @ X3 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_2069_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_2070_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X6: set @ A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X6 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite @ A @ X6 ) ) ) ) ).
% infinite_growing
thf(fact_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ B @ ( F2 @ X4 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_2078_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F2: A > B] :
( ( finite_finite @ 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_2079_finite__transitivity__chain,axiom,
! [A: $tType,A3: set @ A,R: A > A > $o] :
( ( finite_finite @ A @ A3 )
=> ( ! [X3: A] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( R @ X3 @ Y5 ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_2080_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite @ A @ A3 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_2081_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_2082_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2083_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2084_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_2085_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_2086_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_2087_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A3 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_2088_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_2089_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_2090_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_2091_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A3 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2092_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2093_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2094_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_2095_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2096_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B2 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2097_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_2098_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2099_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2100_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_2101_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_2102_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2103_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2104_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_2105_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_2106_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_2107_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_2108_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image2 @ B @ A @ F2 @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_2109_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_2110_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_2111_img__fst,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( member @ A @ A4 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ).
% img_fst
thf(fact_2112_img__snd,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( member @ B @ B3 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ).
% img_snd
thf(fact_2113_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A3: 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 ) @ A3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% pair_in_swap_image
thf(fact_2114_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_2115_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_2116_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_2117_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_2118_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A4 )
= A4 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_2119_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2120_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A4 ) @ ( times_times @ A @ D3 @ B3 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_2121_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_2122_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H2: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2123_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N3: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( inf_inf @ A @ X3 @ Y3 ) )
= ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N3 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H2 @ N3 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2124_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 ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ S ) ) ).
% in_fst_imageE
thf(fact_2125_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 ) )
=> ~ ! [X3: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_2126_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_2127_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ( set_ord_atMost @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( top_top @ A ) ) ) ) ).
% atMost_eq_UNIV_iff
thf(fact_2128_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ( F2 @ Y3 )
= ( F2 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F2 @ A3 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_2129_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: A,X: B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F2 @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_2130_fst__image__mp,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B2: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) @ B2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A3 )
=> ( member @ A @ X @ B2 ) ) ) ).
% fst_image_mp
thf(fact_2131_snd__image__mp,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A3 ) @ B2 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 )
=> ( member @ A @ Y @ B2 ) ) ) ).
% snd_image_mp
thf(fact_2132_in__image__insert__iff,axiom,
! [A: $tType,B2: set @ ( set @ A ),X: A,A3: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B2 )
=> ~ ( member @ A @ X @ C7 ) )
=> ( ( member @ ( set @ A ) @ A3 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X ) @ B2 ) )
= ( ( member @ A @ X @ A3 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2133_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B3 @ A4 )
| ( ord_less_eq @ A @ D3 @ C3 )
| ( ord_less_eq @ A @ B3 @ C3 )
| ( ord_less_eq @ A @ D3 @ A4 ) ) ) ) ).
% Ioc_disjoint
thf(fact_2134_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_2135_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2136_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2137_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A3 )
=> ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2138_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_2139_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N3: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( sup_sup @ A @ X3 @ Y3 ) )
= ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N3 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H2 @ N3 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2140_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_2141_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2142_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2143_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_2144_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2145_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2146_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2147_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_2148_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G2: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2149_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_2150_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_2151_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_2152_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A3 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_2153_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_2154_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ).
% bool_assn_proper(2)
thf(fact_2155_bool__assn__proper_I3_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q )
=> ( proper
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
| ( Q @ H ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_2156_bool__assn__proper_I4_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q )
=> ( proper
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
& ( Q @ H ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_2157_singleton__conv2,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ( ^ [Y4: A,Z5: A] : Y4 = Z5
@ A4 ) )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_2158_singleton__conv,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A4 )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_2159_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_2160_bool__assn__proper_I5_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H )
& ~ ( P @ H ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_2161_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_2162_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) )
= ( ~ ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_2163_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_2164_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_2165_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_2166_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_2167_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_2168_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_2169_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_2170_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_2171_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_2172_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert2 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_2173_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C3: A] :
( ( finite_finite @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C3
@ A3 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_2174_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A3 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min_insert
thf(fact_2175_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_2176_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_2177_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_2178_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 )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_2179_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_2180_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 )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_2181_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% Set.empty_def
thf(fact_2182_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_2183_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_2184_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A4: A] :
( ( ( P @ A4 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A4 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_2185_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A4: A] :
( ( ( P @ A4 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A4 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_2186_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 )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_2187_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,P5: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_2188_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_2189_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_2190_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_2191_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G2 @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2192_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F2 @ S ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_2193_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
@ ^ [I2: nat] : ( G2 @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2194_bot__assn__def,axiom,
( ( bot_bot @ assn )
= ( abs_assn
@ ^ [Uu: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ) ).
% bot_assn_def
thf(fact_2195_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A4: nat,B3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A4 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F2 @ A5 ) )
@ A4
@ B3
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2196_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L ) ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_2197_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A3: set @ A,C3: B] :
( ( member @ A @ X @ A3 )
=> ( ( image2 @ A @ B
@ ^ [X2: A] : C3
@ A3 )
= ( insert2 @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_2198_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C3: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A3 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_2199_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A4: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A4 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_2200_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A4 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_2201_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_2202_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G2 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_2203_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: nat,B3: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A4 @ ( suc @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ A4 @ B3 ) ) @ ( G2 @ B3 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_2204_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G2 @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_2205_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C3 ) @ ( minus_minus @ nat @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_2206_sup__assn__def,axiom,
( ( sup_sup @ assn )
= ( ^ [P3: assn,Q3: assn] :
( abs_assn
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H )
| ( rep_assn @ Q3 @ H ) ) ) ) ) ).
% sup_assn_def
thf(fact_2207_inf__assn__def,axiom,
( ( inf_inf @ assn )
= ( ^ [P3: assn,Q3: assn] :
( abs_assn
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H )
& ( rep_assn @ Q3 @ H ) ) ) ) ) ).
% inf_assn_def
thf(fact_2208_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_2209_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_2210_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_2211_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_2212_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_2213_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_2214_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_2215_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
@ ^ [I2: nat] : ( times_times @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_2216_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_2217_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_2218_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ).
% Min_in
thf(fact_2219_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_2220_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_2221_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: nat,K: nat,G2: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2222_uminus__assn__def,axiom,
( ( uminus_uminus @ assn )
= ( ^ [P3: assn] :
( abs_assn
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H )
& ~ ( rep_assn @ P3 @ H ) ) ) ) ) ).
% uminus_assn_def
thf(fact_2223_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_2224_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G2 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_2225_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_2226_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_2227_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A4 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B3 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A4 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_2228_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A4 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B3 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A4 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_2229_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_2230_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_2231_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A3 )
= M2 )
= ( ( member @ A @ M2 @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M2 @ X2 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_2232_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_2233_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ( member @ A @ M2 @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M2 @ X2 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_2234_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A3 )
=> ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_2235_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_2236_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_2237_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_2238_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_2239_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_2240_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_2241_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_2242_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_2243_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2244_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G2 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2245_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_2246_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_2247_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M4: set @ A,N3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M4 @ N3 )
=> ( ( M4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N3 ) @ ( lattic643756798350308766er_Min @ A @ M4 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_2248_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N3: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ord_min @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N3 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H2 @ N3 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_2249_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_2250_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_2251_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset
thf(fact_2252_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y3 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Min.closed
thf(fact_2253_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A3 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_2254_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( lattic643756798350308766er_Min @ A @ B2 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_2255_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_2256_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_2257_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_2258_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_2259_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G2: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2260_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_2261_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A3 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_2262_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A] :
( ( finite_finite @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A4 = K4 ) @ ( B3 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A4 ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A4 = K4 ) @ ( B3 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2263_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A] :
( ( finite_finite @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A4 ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2264_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A] :
( ~ ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_2265_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C3: A] :
( ( finite_finite @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ C3 )
@ S )
= ( times_times @ A @ ( B3 @ A4 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ C3 )
@ S )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_2266_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_2267_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A3: set @ B,B2: set @ B,F2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
=> ( ( F2 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_2268_atLeast__empty__triv,axiom,
! [A: $tType] :
( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atLeast_empty_triv
thf(fact_2269_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_2270_inf2I,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y: B,B2: A > B > $o] :
( ( A3 @ X @ Y )
=> ( ( B2 @ X @ Y )
=> ( inf_inf @ ( A > B > $o ) @ A3 @ B2 @ X @ Y ) ) ) ).
% inf2I
thf(fact_2271_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A3 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_2272_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_2273_inf2D2,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A3 @ B2 @ X @ Y )
=> ( B2 @ X @ Y ) ) ).
% inf2D2
thf(fact_2274_inf2D1,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A3 @ B2 @ X @ Y )
=> ( A3 @ X @ Y ) ) ).
% inf2D1
thf(fact_2275_inf2E,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,B2: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A3 @ B2 @ X @ Y )
=> ~ ( ( A3 @ X @ Y )
=> ~ ( B2 @ X @ Y ) ) ) ).
% inf2E
thf(fact_2276_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_2277_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_2278_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,A3: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
!= ( one_one @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_2279_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_2280_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G2 @ X2 ) @ ( H2 @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_2281_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_ge_1
thf(fact_2282_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,X: B > A,Y: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I4 )
& ( ( X @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I4 )
& ( ( Y @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I4 )
& ( ( times_times @ A @ ( X @ I2 ) @ ( Y @ I2 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2283_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G2 @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_2284_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2285_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G2: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X12: A,Y12: A,X23: A,Y23: A] :
( ( ( R @ X12 @ X23 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times @ A @ X12 @ Y12 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2286_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A3 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_2287_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T4: set @ C,S: set @ B,I: C > B,J: B > C,T5: set @ C,G2: B > A,H2: C > A] :
( ( finite_finite @ B @ S6 )
=> ( ( finite_finite @ C @ T4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S @ S6 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S6 )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T4 )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2288_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A,B2: set @ B] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B2 ) @ ( G2 @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_restrict
thf(fact_2289_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X2: B] :
( ( G2 @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2290_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,I: A,F2: A > B] :
( ( finite_finite @ A @ I4 )
=> ( ( member @ A @ I @ I4 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I4 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2291_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,F2: A > B] :
( ( finite_finite @ A @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2292_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A3: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A3 )
=> ( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2293_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ B,A3: set @ B,B2: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C2 @ A3 ) )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2294_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ B,A3: set @ B,B2: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C2 @ A3 ) )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C2 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2295_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2296_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2297_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S: set @ B,H2: B > A,G2: B > A] :
( ( finite_finite @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2298_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2299_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_2300_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A,B2: set @ B] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_2301_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S: set @ B,H2: B > A,G2: B > A] :
( ( finite_finite @ B @ T5 )
=> ( ( finite_finite @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( H2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T5 ) )
=> ( ( G2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S @ T5 ) )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_2302_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G2 @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_2303_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less @ A @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2304_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2305_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G2: B > A,X: B] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert2 @ B @ X @ A3 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2306_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_2307_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_2308_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ B2 @ A3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_2309_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C3: B > A] :
( ( finite_finite @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C3 @ K4 ) )
@ S )
= ( times_times @ A @ ( B3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C3 @ K4 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_2310_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B2: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B5 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B2 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2311_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_2312_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F2: B > A,A4: B] :
( ( finite_finite @ B @ A3 )
=> ( ( ( F2 @ A4 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A4 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( F2 @ A4 ) ) ) )
& ( ~ ( member @ B @ A4 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_2313_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
@ ^ [I2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_2314_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
@ ^ [I2: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_2315_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A4 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A4 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2316_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_2317_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2318_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
@ N2
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_2319_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_2320_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y
@ A3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_2321_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o,F2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F2 @ X2 ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_2322_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A,P: B > $o] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_2323_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C3: nat > A] :
( ( ( ( finite_finite @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A3 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2324_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C3: nat > A,D3: nat > A] :
( ( ( ( finite_finite @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D3 @ I2 ) )
@ A3 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D3 @ I2 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2325_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F2: B > A,A3: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ R3 @ ( F2 @ N2 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_2326_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,A3: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ ( F2 @ N2 ) @ R3 )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_2327_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A3: set @ A,G2: C > B,B2: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I2 ) @ ( G2 @ J3 ) )
@ B2 )
@ A3 ) ) ) ).
% sum_product
thf(fact_2328_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F2: B > A,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2329_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,I4: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I2 ) )
@ I4 )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I4 ) ) ) ) ).
% sum_power_add
thf(fact_2330_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I4: set @ B,F2: B > A] :
( ( finite_finite @ B @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I4 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2331_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_2332_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K5: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ K5 @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_2333_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G2: B > A,X: B] :
( ( finite_finite @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert2 @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2334_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A3 )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2335_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A4: B,F2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( ( member @ B @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( F2 @ A4 ) ) ) )
& ( ~ ( member @ B @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2336_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B2: set @ B,G2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_2337_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C3: B > A] :
( ( finite_finite @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C3 @ K4 ) )
@ S )
= ( plus_plus @ A @ ( B3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C3 @ K4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2338_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ K4 ) ) @ K4 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2339_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A3: set @ C,F2: C > B] :
( ( member @ C @ I @ A3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ( finite_finite @ C @ A3 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2340_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F2: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_2341_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less @ A @ ( F2 @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_2342_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,X: A > B,A4: A > B,B3: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I4 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A4 @ I3 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( times_times @ B @ ( A4 @ I2 ) @ ( X @ I2 ) )
@ I4 )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2343_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_finite @ A @ S )
=> ( ( finite_finite @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G2 @ S ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F2 @ ( G2 @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y2: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
& ( ( G2 @ X2 )
= Y2 ) ) ) ) )
@ ( F2 @ Y2 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_2344_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_2345_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2346_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
@ ^ [Q8: nat] : ( ord_less @ nat @ Q8 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2347_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2348_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A4 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_2349_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A4 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( power_power @ A @ A4 @ K4 ) ) @ ( power_power @ A @ B3 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_2350_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A4 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A4 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B3 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2351_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_2352_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A4: 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 @ M2 ) @ A4 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A4 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2353_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ 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 ) ) @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_2354_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_2355_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_2356_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A4: 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 @ M2 ) @ A4 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( 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 ) @ A4 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2357_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2358_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_2359_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_2360_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ 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 ) ) @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2361_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ 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 @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2362_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2363_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_2364_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M2: 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 ) @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R3 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2365_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R2: A] : ( product_Pair @ A @ A @ Q8 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2366_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R2: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R2 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q8 ) @ R2 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_2367_normalize__def,axiom,
( normalize
= ( ^ [P7: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P7 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P7 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_2368_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A6: A,B5: A,X3: A] :
( ( member @ A @ A6 @ S )
=> ( ( member @ A @ B5 @ S )
=> ( ( ord_less_eq @ A @ A6 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B5 )
=> ( member @ A @ X3 @ S ) ) ) ) )
=> ? [A6: A,B5: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B5 ) )
| ( S
= ( set_ord_atMost @ A @ B5 ) )
| ( S
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S
= ( set_ord_atLeast @ A @ A6 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A6 @ B5 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A6 @ B5 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A6 @ B5 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_2369_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q8: int,R2: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R2 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R2 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q8 ) @ R2 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2370_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_finite @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I4 )
& ( ( F2 @ I2 )
!= ( 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_2371_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A] :
( ( gcd_gcd @ A @ A4 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_2372_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A4 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_2373_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2374_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A4: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( F2 @ A4 @ B3 ) ) ).
% case_prod_conv
thf(fact_2375_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A4 @ B3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_2376_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_2377_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_2378_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A4: A,B3: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ A3 )
=> ( member @ C @ ( F2 @ A4 @ B3 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_2379_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_2380_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,B3: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ).
% Gcd_2
thf(fact_2381_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R2: A] : ( product_Pair @ A @ A @ Q8 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R2 ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2382_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_2383_gcd_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( semigroup @ A @ ( gcd_gcd @ A ) ) ) ).
% gcd.semigroup_axioms
thf(fact_2384_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2385_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G2: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F2 @ X3 @ Y3 )
= ( G2 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_2386_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y2: B] : ( F2 @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_2387_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_2388_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,K: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K @ M2 ) @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add_mult
thf(fact_2389_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ ( times_times @ A @ K @ B3 ) ) ) ).
% gcd_dvd_prod
thf(fact_2390_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2391_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_2392_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B3 @ A4 ) @ C3 )
= ( gcd_gcd @ A @ B3 @ C3 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_2393_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B3 @ ( times_times @ A @ C3 @ A4 ) )
= ( gcd_gcd @ A @ B3 @ C3 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_2394_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B3 @ ( divide_divide @ A @ C3 @ A4 ) )
= ( gcd_gcd @ A @ B3 @ C3 ) ) ) ) ).
% gcd_div_unit2
thf(fact_2395_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B3 @ A4 ) @ C3 )
= ( gcd_gcd @ A @ B3 @ C3 ) ) ) ) ).
% gcd_div_unit1
thf(fact_2396_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_2397_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_2398_rat__abs__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_abs_code
thf(fact_2399_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_2400_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_2401_sum__diff1__nat,axiom,
! [A: $tType,A4: A,A3: set @ A,F2: A > nat] :
( ( ( member @ A @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( F2 @ A4 ) ) ) )
& ( ~ ( member @ A @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ) ).
% sum_diff1_nat
thf(fact_2402_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( gcd_gcd @ A @ A4 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_2403_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( gcd_gcd @ A @ A4 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_2404_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
@ ^ [I2: nat] : ( G2 @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_2405_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_2406_rat__uminus__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_uminus_code
thf(fact_2407_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_2408_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_2409_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_2410_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_2411_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
@ ^ [I2: nat] : ( G2 @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_2412_rat__minus__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P5 @ 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 @ P5 ) ) ) ).
% rat_minus_code
thf(fact_2413_rat__divide__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P5 @ 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 @ P5 ) ) ) ).
% rat_divide_code
thf(fact_2414_rat__times__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P5 @ 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 @ P5 ) ) ) ).
% rat_times_code
thf(fact_2415_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_2416_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
@ ^ [P7: nat] : ( times_times @ A @ ( power_power @ A @ X @ P7 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P7 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2417_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
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2418_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q8: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q8 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2419_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_2420_rat__plus__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P5 @ 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 @ P5 ) ) ) ).
% rat_plus_code
thf(fact_2421_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
@ ^ [I2: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2422_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I4: set @ A,F2: A > B,I: A] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I4 )
& ( ( F2 @ I2 )
!= ( 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_2423_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q8: nat,R2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R2 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R2 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q8 ) @ R2 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2424_rat__inverse__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
= ( 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 @ P5 ) ) ) ).
% rat_inverse_code
thf(fact_2425_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
= P5 ) ).
% case_prod_Pair_iden
thf(fact_2426_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q8: code_integer,R2: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R2 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R2 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q8 ) @ R2 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_2427_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_2428_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) ) ).
% less_by_empty
thf(fact_2429_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_2430_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F2: A > B > C,G2: A > B > C,P5: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q4 )
=> ( ( F2 @ X3 @ Y3 )
= ( G2 @ X3 @ Y3 ) ) )
=> ( ( P5 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P5 )
= ( product_case_prod @ A @ B @ C @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_2431_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B3: B] :
( ( F2 @ A4 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ).
% case_prodI
thf(fact_2432_case__prodI2,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,C3: A > B > $o] :
( ! [A6: A,B5: B] :
( ( P5
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( C3 @ A6 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P5 ) ) ).
% case_prodI2
thf(fact_2433_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),A4: B,B3: C] :
( ( member @ A @ Z2 @ ( C3 @ A4 @ B3 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A4 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_2434_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C3: A > B > ( set @ C )] :
( ! [A6: A,B5: B] :
( ( P5
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( member @ C @ Z2 @ ( C3 @ A6 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P5 ) ) ) ).
% mem_case_prodI2
thf(fact_2435_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
( ! [A6: A,B5: B] :
( ( ( product_Pair @ A @ B @ A6 @ B5 )
= P5 )
=> ( C3 @ A6 @ B5 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P5 @ X ) ) ).
% case_prodI2'
thf(fact_2436_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_2437_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
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ R )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_2438_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M: num,N2: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N2 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_2439_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_2440_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_2441_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A,Q: A > $o] :
( ( P @ A4 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_2442_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_2443_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
@ ^ [Y4: num,Z5: num] : Y4 = Z5
@ R
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_2444_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_2445_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P5: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P5 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P5
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_2446_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( F2 @ A4 @ B3 ) ) ).
% case_prodD
thf(fact_2447_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P5 )
=> ~ ! [X3: A,Y3: B] :
( ( P5
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_2448_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
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ R
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_2449_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A4: A,B3: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A4 @ B3 ) @ C3 )
=> ( R @ A4 @ B3 @ C3 ) ) ).
% case_prodD'
thf(fact_2450_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P5 @ Z2 )
=> ~ ! [X3: A,Y3: B] :
( ( P5
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_2451_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A6 @ B10 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_2452_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A,Q: A > $o] :
( ( P @ A4 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A6 @ B10 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_2453_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_2454_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_2455_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya: A] :
( ( P @ Ya )
=> ( ord_less_eq @ A @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_2456_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya: A] :
( ( P @ Ya )
=> ( ord_less_eq @ A @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_2457_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_2458_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_2459_Gcd__in,axiom,
! [A3: set @ nat] :
( ! [A6: nat,B5: nat] :
( ( member @ nat @ A6 @ A3 )
=> ( ( member @ nat @ B5 @ A3 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A6 @ B5 ) @ A3 ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ).
% Gcd_in
thf(fact_2460_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_2461_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_2462_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
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ R
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_2463_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 )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y2 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_2464_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 )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_2465_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y2 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V2 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_2466_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,P5: B > A,I: B] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I4 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) )
& ( ~ ( member @ B @ I @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
= ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_2467_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_2468_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y3: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_2469_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ I4 )
& ( ( G2 @ X2 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ I4 ) ) ) ).
% prod.non_neutral'
thf(fact_2470_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ 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_2471_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
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5
@ R
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_2472_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 )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) ) ) ).
% times_rat.rsp
thf(fact_2473_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y2 @ V2 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_2474_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V2 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_2475_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 )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y2 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y2 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) ) ) ).
% plus_rat.rsp
thf(fact_2476_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y2 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V2 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_2477_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite @ B @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G2 @ I2 ) @ ( H2 @ I2 ) )
@ I4 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G2 @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I4 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_2478_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T5: set @ B,G2: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ T5 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_2479_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T5: set @ B,H2: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( H2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_2480_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T5: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ T5 )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_2481_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T5: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G2 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G2 @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_2482_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_2483_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_2484_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I4 )
& ( ( G2 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I4 )
& ( ( H2 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G2 @ I2 ) @ ( H2 @ I2 ) )
@ I4 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G2 @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I4 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_2485_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P7: B > A,I5: set @ B] :
( if @ A
@ ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P7 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P7
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P7 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_2486_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ pcr_int
@ ^ [N2: nat] : ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_2487_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 )
@ ^ [X2: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X2 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_2488_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 )
@ ^ [X2: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_2489_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_2490_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
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ).
% uminus_rat.rsp
thf(fact_2491_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y2 @ V2 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_2492_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V2 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_2493_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
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_2494_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_2495_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_2496_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y2 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V2 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y2 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V2 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_2497_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R: C > assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X2: C] : ( rep_assn @ ( R @ X2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_2498_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I4: set @ A,A3: A > ( set @ B )] :
( ( finite_finite @ A @ I4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I4 )
=> ( finite_finite @ B @ ( A3 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I4 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ I4 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] : ( finite_card @ B @ ( A3 @ I2 ) )
@ I4 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_2499_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite @ 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_2500_ex__assn__const,axiom,
! [A: $tType,C3: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : C3 )
= C3 ) ).
% ex_assn_const
thf(fact_2501_mod__ex__dist,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
= ( ? [X2: A] : ( rep_assn @ ( P @ X2 ) @ H2 ) ) ) ).
% mod_ex_dist
thf(fact_2502_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_2503_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_2504_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_2505_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_2506_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A3 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_2507_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_2508_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_2509_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_2510_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C3: A] :
( ( finite_finite @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C3
@ A3 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_2511_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite @ 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_2512_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic643756798349783984er_Max @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_2513_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Max_Sup
thf(fact_2514_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A4: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A4 ) )
=> ( ! [Y3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) )
=> ( ord_less_eq @ A @ A4 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A4 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_2515_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_2516_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X6: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_2517_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X6 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_2518_insert__partition,axiom,
! [A: $tType,X: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F5 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( insert2 @ ( set @ A ) @ X @ F5 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert2 @ ( set @ A ) @ X @ F5 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_2519_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ).
% Max_in
thf(fact_2520_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V: A] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A] :
( ( rep_assn @ ( P @ X3 ) @ H3 )
=> ( X3 = V ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V ) ) ) ).
% ex_one_point_gen
thf(fact_2521_mod__exI,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X4: A] : ( rep_assn @ ( P @ X4 ) @ H2 )
=> ( rep_assn @ ( ex_assn @ A @ P ) @ H2 ) ) ).
% mod_exI
thf(fact_2522_mod__exE,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
=> ~ ! [X3: A] :
~ ( rep_assn @ ( P @ X3 ) @ H2 ) ) ).
% mod_exE
thf(fact_2523_ent__ex__preI,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ! [X3: A] : ( entails @ ( P @ X3 ) @ Q )
=> ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ent_ex_preI
thf(fact_2524_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: A > assn,X: A] :
( ( entails @ P @ ( Q @ X ) )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).
% ent_ex_postI
thf(fact_2525_ex__distrib__star,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( times_times @ assn @ ( P @ X2 ) @ Q ) )
= ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_star
thf(fact_2526_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_2527_ex__distrib__or,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( sup_sup @ assn @ ( P @ X2 ) @ Q ) )
= ( sup_sup @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_or
thf(fact_2528_ex__join__or,axiom,
! [A: $tType,P: A > assn,Q: A > assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( sup_sup @ assn @ ( P @ X2 ) @ ( ex_assn @ A @ Q ) ) )
= ( ex_assn @ A
@ ^ [X2: A] : ( sup_sup @ assn @ ( P @ X2 ) @ ( Q @ X2 ) ) ) ) ).
% ex_join_or
thf(fact_2529_ex__distrib__and,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( inf_inf @ assn @ ( P @ X2 ) @ Q ) )
= ( inf_inf @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_and
thf(fact_2530_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,M4: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ M4 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ M4 ) ) ) ) ).
% cSUP_least
thf(fact_2531_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A4: A] :
( ( finite_finite @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X6 ) @ A4 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X6 )
=> ( ord_less @ A @ X2 @ A4 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_2532_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_2533_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A4 ) ) ) ) ).
% cSup_abs_le
thf(fact_2534_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_2535_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic5882676163264333800up_fin @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_2536_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M2 )
= ( ( member @ A @ M2 @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M2 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_2537_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_2538_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M2 @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M2 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_2539_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A3 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_2540_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_2541_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_2542_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 )
@ ^ [X2: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X2 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X2 ) ) ) ).
% uminus_int.rsp
thf(fact_2543_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_2544_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_2545_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_2546_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M4: set @ A,N3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M4 @ N3 )
=> ( ( M4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M4 ) @ ( lattic643756798349783984er_Max @ A @ N3 ) ) ) ) ) ) ).
% Max_mono
thf(fact_2547_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F2 @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_2548_card__partition,axiom,
! [A: $tType,C2: set @ ( set @ A ),K: nat] :
( ( finite_finite @ ( set @ A ) @ C2 )
=> ( ( finite_finite @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ( ! [C4: set @ A] :
( ( member @ ( set @ A ) @ C4 @ C2 )
=> ( ( finite_card @ A @ C4 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C2 )
=> ( ( member @ ( set @ A ) @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C2 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_2549_dvd__partition,axiom,
! [A: $tType,C2: set @ ( set @ A ),K: nat] :
( ( finite_finite @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X3 ) ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C2 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_2550_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,A3: B > ( set @ C ),G2: C > A] :
( ( finite_finite @ B @ I4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( finite_finite @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I4 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( A3 @ X2 ) )
@ I4 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_2551_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,A3: B > ( set @ C ),G2: C > A] :
( ( finite_finite @ B @ I4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( finite_finite @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I4 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( A3 @ X2 ) )
@ I4 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_2552_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V2 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V2 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_2553_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y2 @ V2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y2 @ V2 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_2554_UN__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C2: set @ F3,A3: set @ E,B2: F3 > ( set @ E )] :
( ( ( C2
= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2555_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C2: set @ C,A3: C > ( set @ D ),B2: set @ D] :
( ( ( C2
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C2 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_2556_UN__singleton,axiom,
! [A: $tType,A3: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X2: A] : ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_2557_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A4: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A4 @ ( B2 @ X2 ) )
@ C2 ) )
= ( insert2 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2558_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A3 )
& ( ord_less @ A @ X2 @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_2559_UN__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C3: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y2: B] : C3
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y2: B] : C3
@ A3 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_2560_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_2561_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_2562_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_2563_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( ord_less @ A @ X2 @ Y2 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_2564_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_2565_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_2566_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_insert
thf(fact_2567_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A3: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B2 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_2568_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B2 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_2569_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_2570_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_2571_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 )
@ ^ [I2: set @ ( product_prod @ A @ B ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ I2 )
@ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2572_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 )
@ ^ [I2: C,X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( R3 @ I2 ) )
@ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_2573_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X7: set @ nat] :
( if @ nat
@ ( X7
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X7 ) ) ) ) ).
% Sup_nat_def
thf(fact_2574_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( 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 ) @ S7 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2575_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_2576_Union__empty__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A3 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_2577_empty__Union__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A3 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_2578_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: $o > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ $o @ A @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_2579_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_2580_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,X: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I4 )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_2581_Sup__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_union_distrib
thf(fact_2582_Union__disjoint,axiom,
! [A: $tType,C2: set @ ( set @ A ),A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_2583_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I4: set @ B,A3: B > A] :
( ( member @ B @ K @ I4 )
=> ( ( sup_sup @ A @ ( A3 @ K ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A3 @ I4 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A3 @ I4 ) ) ) ) ) ).
% SUP_absorb
thf(fact_2584_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,G2: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A3 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_2585_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B2 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_2586_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B2 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_2587_UN__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_2588_UN__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( bot_bot @ ( set @ A ) )
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_2589_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,C3: A,F2: B > A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I4 )
=> ( ord_less_eq @ A @ C3 @ ( F2 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= C3 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( ( F2 @ X2 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2590_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_2591_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C3: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y2: B] : C3
@ A3 ) )
= ( bot_bot @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y2: B] : C3
@ A3 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_2592_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: B,A3: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( sup_sup @ A @ ( F2 @ A4 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% SUP_insert
thf(fact_2593_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M4: B > A,A3: set @ B,B2: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M4 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M4 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M4 @ B2 ) ) ) ) ) ).
% SUP_union
thf(fact_2594_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A4: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A4 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2595_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C2: set @ C,A3: C > ( set @ D ),B2: set @ D] :
( ( ( C2
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C2 ) ) @ B2 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_2596_UN__extend__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C2: set @ F3,A3: set @ E,B2: F3 > ( set @ E )] :
( ( ( C2
= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B2 @ C2 ) ) )
= A3 ) )
& ( ( C2
!= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B2 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_2597_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ ( F2 @ X2 ) @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= ( image2 @ B @ A @ F2 @ A3 ) ) ).
% UNION_singleton_eq_range
thf(fact_2598_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_2599_Union__image__empty,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A3 ) ).
% Union_image_empty
thf(fact_2600_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 )
@ ^ [R2: code_integer,S5: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R2 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ S5 ) )
@ ( S5
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_2601_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,C3: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X2: B] : C3
@ A3 ) )
= C3 ) ) ).
% subset_mset.cSUP_const
thf(fact_2602_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_finite_insert
thf(fact_2603_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_2604_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_2605_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_2606_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert2 @ $o @ $false @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_2607_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_2608_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_2609_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_2610_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 )
@ ^ [R2: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S5 ) ) )
@ ( 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 )
@ ^ [R2: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S5 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_2611_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A4: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B2 ) @ A4 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( ( inf_inf @ A @ X2 @ A4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_2612_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y2: A] :
( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y2 ) )
| ( ( ord_less_eq @ nat @ ( F @ X2 ) @ ( F @ Y2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_2613_Gcd__eq__Max,axiom,
! [M4: set @ nat] :
( ( finite_finite @ 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 )
@ ^ [M: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) )
@ M4 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_2614_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A3: B > ( set @ A ),I: B,B2: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A3 @ I @ B2 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert2 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_2615_fold__union__pair,axiom,
! [B: $tType,A: $tType,B2: set @ A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( finite_finite @ A @ B2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y2: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B2 ) )
@ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y2: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y2 ) )
@ A3
@ B2 ) ) ) ).
% fold_union_pair
thf(fact_2616_Inf__top__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( X2
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_2617_Inf__top__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( X2
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_2618_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_2619_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_2620_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( ord_less @ A @ Y2 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_2621_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_2622_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_2623_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_2624_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_2625_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_2626_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : C3
@ A3 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_2627_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( top_top @ A )
@ A3 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_2628_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B2 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B2 @ X2 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_2629_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A3: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B2 @ X2 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_2630_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A3 )
& ( ord_less @ A @ ( F2 @ Y2 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_2631_INT__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C3: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y2: B] : C3
@ A3 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y2: B] : C3
@ A3 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_2632_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_2633_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C2: set @ D,A3: set @ C,B2: D > ( set @ C )] :
( ( ( C2
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2634_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ A,A3: A > ( set @ B ),B2: set @ B] :
( ( ( C2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C2 ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_2635_INT__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C2: set @ E,A3: E > ( set @ F3 ),B2: set @ F3] :
( ( ( C2
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) )
= ( top_top @ ( set @ F3 ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) )
= ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A3 @ C2 ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_2636_INT__simps_I4_J,axiom,
! [G3: $tType,H8: $tType,C2: set @ H8,A3: set @ G3,B2: H8 > ( set @ G3 )] :
( ( ( C2
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( top_top @ ( set @ G3 ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) )
= ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B2 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2637_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: A,B2: set @ A] :
( ( sup_sup @ A @ A4 @ ( complete_Inf_Inf @ A @ B2 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A4 ) @ B2 ) ) ) ) ).
% sup_Inf
thf(fact_2638_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A4: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B2 ) @ A4 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( ( sup_sup @ A @ X2 @ A4 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_2639_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( complete_Inf_Inf @ A @ A3 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A3 ) ) ) ) ).
% Inf_fold_inf
thf(fact_2640_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B2: set @ B,A4: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B2 ) ) @ A4 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B4: B] : ( sup_sup @ A @ ( F2 @ B4 ) @ A4 )
@ B2 ) ) ) ) ).
% INF_sup
thf(fact_2641_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A4: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B2 ) @ A4 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B4: A] : ( sup_sup @ A @ B4 @ A4 )
@ B2 ) ) ) ) ).
% Inf_sup
thf(fact_2642_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: A,F2: B > A,B2: set @ B] :
( ( sup_sup @ A @ A4 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B2 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B4: B] : ( sup_sup @ A @ A4 @ ( F2 @ B4 ) )
@ B2 ) ) ) ) ).
% sup_INF
thf(fact_2643_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A3: set @ B,G2: C > A,B2: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) )
= ( 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 ) )
@ B2 ) )
@ A3 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_2644_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_2645_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ).
% cInf_greatest
thf(fact_2646_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A4: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ A4 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ord_less_eq @ A @ Y3 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A4 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_2647_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Z2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ X3 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_2648_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,X: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I4 )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_2649_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_2650_Inter__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B2: set @ A] :
( ! [X8: set @ A] :
( ( member @ ( set @ A ) @ X8 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ B2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_2651_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_2652_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_2653_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F2: B > A,C3: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) )
= C3 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( ( F2 @ X2 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2654_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M2: A,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ M2 @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_2655_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A4: A] :
( ( finite_finite @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A4 @ ( complete_Inf_Inf @ A @ X6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X6 )
=> ( ord_less @ A @ A4 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_2656_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Inf_le_Sup
thf(fact_2657_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_2658_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S ) ) @ A4 ) ) ) ) ).
% cInf_abs_ge
thf(fact_2659_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_2660_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_2661_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_2662_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic643756798350308766er_Min @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_2663_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Min_Inf
thf(fact_2664_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: A] :
( ( finite_finite @ A @ A3 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B2 @ A3 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_2665_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic7752659483105999362nf_fin @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_2666_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_2667_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_2668_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_2669_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C3: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y2: B] : C3
@ A3 ) )
= ( top_top @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y2: B] : C3
@ A3 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_2670_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
@ ^ [I2: B] : ( inf_inf @ A @ ( F2 @ I2 ) @ X )
@ I4 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_2671_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
@ ^ [I2: B] : ( inf_inf @ A @ X @ ( F2 @ I2 ) )
@ I4 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I4 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_2672_INT__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_2673_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C2: set @ A,A3: A > ( set @ B ),B2: set @ B] :
( ( ( C2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C2 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_2674_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C2: set @ D,A3: set @ C,B2: D > ( set @ C )] :
( ( ( C2
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C2 ) ) )
= A3 ) )
& ( ( C2
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_2675_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A3: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= A3 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B11 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_2676_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A3: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A3 )
= A3 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A3 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A3 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_2677_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_2678_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_2679_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite @ 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_2680_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( complete_Sup_Sup @ A @ A3 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% Sup_fold_sup
thf(fact_2681_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A3 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_2682_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite @ A @ A3 )
=> ( ( image2 @ A @ B @ F2 @ A3 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K4: A] : ( insert2 @ B @ ( F2 @ K4 ) )
@ ( bot_bot @ ( set @ B ) )
@ A3 ) ) ) ).
% image_fold_insert
thf(fact_2683_INT__extend__simps_I3_J,axiom,
! [F3: $tType,E: $tType,C2: set @ E,A3: E > ( set @ F3 ),B2: set @ F3] :
( ( ( C2
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A3 @ C2 ) ) @ B2 )
= ( minus_minus @ ( set @ F3 ) @ ( top_top @ ( set @ F3 ) ) @ B2 ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A3 @ C2 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A3 @ X2 ) @ B2 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2684_INT__extend__simps_I4_J,axiom,
! [G3: $tType,H8: $tType,C2: set @ H8,A3: set @ G3,B2: H8 > ( set @ G3 )] :
( ( ( C2
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B2 @ C2 ) ) )
= A3 ) )
& ( ( C2
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B2 @ X2 ) )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_2685_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A8: set @ A] : ( if @ A @ ( finite_finite @ A @ A8 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A8 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_2686_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A3: set @ B,F2: B > A,Y: A] :
( ( ( member @ B @ X @ A3 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y ) @ A3 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y ) @ A3 )
= ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_2687_card__UNION,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite @ ( set @ A ) @ A3 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ( finite_finite @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( 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 @ A3 )
& ( I5
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_2688_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_2689_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_2690_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_2691_Set__filter__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite @ A @ A3 )
=> ( ( filter3 @ A @ P @ A3 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X2: A,A11: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A11 ) @ A11 )
@ ( bot_bot @ ( set @ A ) )
@ A3 ) ) ) ).
% Set_filter_fold
thf(fact_2692_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_2693_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,C3: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X2: B] : C3
@ A3 ) )
= C3 ) ) ).
% subset_mset.cINF_const
thf(fact_2694_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_2695_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 )
@ ^ [I2: set @ ( product_prod @ A @ B ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ I2 )
@ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% INF_Int_eq2
thf(fact_2696_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 )
@ ^ [I2: C,X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( R3 @ I2 ) )
@ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_2697_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_2698_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B2: set @ I6,F5: I6 > ( filter @ A )] :
( ! [X8: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X8 @ B2 )
=> ( ( finite_finite @ 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 @ B2 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_2699_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( 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 ) @ S7 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2700_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_2701_in__finite__psubset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B2 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
& ( finite_finite @ A @ B2 ) ) ) ).
% in_finite_psubset
thf(fact_2702_Id__on__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( id_on @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X2: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A3 ) ) ) ).
% Id_on_fold
thf(fact_2703_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X2: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A8 ) ) ) ) ).
% Id_on_def
thf(fact_2704_Pow__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( pow @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A8 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A3 ) ) ) ).
% Pow_fold
thf(fact_2705_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A3: set @ A,B2: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ S )
=> ( ( finite_finite @ A @ A3 )
=> ( ( finite_finite @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( finite_fold @ A @ B @ F2 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) @ B2 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_2706_Id__onI,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( id_on @ A @ A3 ) ) ) ).
% Id_onI
thf(fact_2707_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_2708_Pow__singleton__iff,axiom,
! [A: $tType,X6: set @ A,Y6: set @ A] :
( ( ( pow @ A @ X6 )
= ( insert2 @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_2709_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_2710_Inf__filter__not__bot,axiom,
! [A: $tType,B2: set @ ( filter @ A )] :
( ! [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B2 )
=> ( ( finite_finite @ ( filter @ A ) @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X8 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B2 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_2711_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I4: set @ A,F5: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I4 )
=> ? [X4: A] :
( ( member @ A @ X4 @ I4 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X4 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I3 ) @ ( F5 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I4 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ I4 )
& ( ( F5 @ X2 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_2712_Pow__bottom,axiom,
! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% Pow_bottom
thf(fact_2713_Pow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( pow @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_2714_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A3 ) )
= ( ( X = Y )
& ( member @ A @ X @ A3 ) ) ) ).
% Id_on_iff
thf(fact_2715_Id__on__eqI,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ A] :
( ( A4 = B3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( id_on @ A @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_2716_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( C3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_2717_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A3: set @ A,X: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ A3 )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_2718_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ S )
=> ( ( finite_finite @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_2719_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_2720_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_2721_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 )
@ ^ [R2: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S5 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_2722_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_2723_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_2724_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B2: set @ A] :
( ( finite_finite @ A @ B2 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X2: B,Z3: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y2: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) )
@ Z3
@ B2 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_2725_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_2726_total__onI,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) )
=> ( total_on @ A @ A3 @ R3 ) ) ).
% total_onI
thf(fact_2727_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A8 )
=> ( ( X2 != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_2728_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_2729_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y2: B,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R6 ) )
=> ( Y2 = Z3 ) )
& ! [X2: A,Y2: A,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y2 @ Z3 ) @ R6 ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% bijective_def
thf(fact_2730_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ B )] :
( ( finite_finite @ ( 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 ) ) )
@ ^ [X2: C,Y2: A,A8: 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,Z3: B,A11: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y2 = W3 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X2 @ Z3 ) @ A11 ) @ A11 ) )
@ A8
@ S ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_2731_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,H2: B > C,G2: C > A] :
( ( finite_finite @ B @ A3 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( member @ B @ Y3 @ A3 )
=> ( ( X3 != Y3 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y3 ) )
=> ( ( G2 @ ( H2 @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( image2 @ B @ C @ H2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G2 @ H2 ) @ A3 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_2732_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_2733_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B2: A,F2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( sup_sup @ A @ B2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ B2 @ A3 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_2734_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% SUP_fold_sup
thf(fact_2735_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite @ B @ A3 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ ( top_top @ A ) @ A3 ) ) ) ) ).
% INF_fold_inf
thf(fact_2736_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F2: C > A,P5: product_prod @ C @ B] :
( ( Q4
= ( product_apfst @ C @ A @ B @ F2 @ P5 ) )
=> ~ ! [X3: C,Y3: B] :
( ( P5
= ( product_Pair @ C @ B @ X3 @ Y3 ) )
=> ( Q4
!= ( product_Pair @ A @ B @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_2737_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X6: set @ ( product_prod @ C @ B )] :
( ( finite_finite @ ( 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 ) @ X6 )
= ( 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,Z3: 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 ) @ Z3 ) @ A11 )
@ A11 ) )
@ X6
@ S ) ) ) ).
% insert_relcomp_union_fold
thf(fact_2738_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: A > B > B,A3: set @ A,Z2: B,Y: B,A4: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A3 @ Y )
=> ( ( member @ A @ A4 @ A3 )
=> ? [Y7: B] :
( ( Y
= ( F2 @ A4 @ Y7 ) )
& ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y7 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_2739_Field__insert,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) ) ) ).
% Field_insert
thf(fact_2740_max__ext_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y6: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ A @ X6 )
=> ( ( finite_finite @ A @ Y6 )
=> ( ( Y6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y6 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2741_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty1
thf(fact_2742_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] :
( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty2
thf(fact_2743_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_2744_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_2745_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A4: A,C3: B,R3: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ C3 ) @ ( relcomp @ A @ C @ B @ R3 @ S2 ) )
=> ~ ! [B5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A4 @ B5 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C3 ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_2746_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R3: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R3 @ S2 ) )
=> ~ ! [X3: A,Y3: C,Z4: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z4 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z4 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_2747_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A4: A,B3: B,R3: set @ ( product_prod @ A @ B ),C3: C,S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C3 ) @ S2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A4 @ C3 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_2748_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) )
= ( ? [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 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_2749_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) )
=> ~ ! [B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B5 ) @ R3 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_2750_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_2751_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_2752_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_2753_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_2754_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_2755_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_2756_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_2757_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_2758_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_2759_union__comp__emptyL,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),C2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A3 @ C2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B2 @ C2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) @ C2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_2760_union__comp__emptyR,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A ),C2: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A3 @ C2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A3 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B2 @ C2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_2761_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F: A > B > B,Z3: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z3 ) )
| ? [X2: A,A8: set @ A,Y2: B] :
( ( A12
= ( insert2 @ A @ X2 @ A8 ) )
& ( A23
= ( F @ X2 @ Y2 ) )
& ~ ( member @ A @ X2 @ A8 )
& ( finite_fold_graph @ A @ B @ F @ Z3 @ A8 @ Y2 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_2762_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 ) )
=> ~ ! [X3: A,A9: set @ A] :
( ( A1
= ( insert2 @ A @ X3 @ A9 ) )
=> ! [Y3: B] :
( ( A22
= ( F2 @ X3 @ Y3 ) )
=> ( ~ ( member @ A @ X3 @ A9 )
=> ~ ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A9 @ Y3 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_2763_max__ext__additive,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,R: set @ ( product_prod @ A @ A ),C2: set @ A,D4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B2 ) @ ( max_ext @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C2 @ D4 ) @ ( max_ext @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ D4 ) ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_ext_additive
thf(fact_2764_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( finite_finite @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite @ ( 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 ) ) )
@ ^ [X2: A,Y2: B,A8: 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,Z3: C,A11: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y2 = W3 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Z3 ) @ A11 ) @ A11 ) )
@ A8
@ S ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_2765_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B2 )
=> ( finite_finite @ B @ X3 ) )
=> ( ! [A13: set @ B] :
( ( member @ ( set @ B ) @ A13 @ B2 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B2 )
=> ( ( A13 != A24 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A13 )
=> ( ( member @ B @ X3 @ A24 )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ B2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G2 @ B2 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_2766_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 )
@ ^ [X2: A,Y2: B] : ( product_Pair @ B @ A @ Y2 @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_2767_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 )
@ ^ [X2: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_2768_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite @ ( 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,Z3: 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 ) @ Z3 ) @ A11 )
@ A11 ) )
@ ( relcomp @ C @ A @ B @ R @ S )
@ S ) ) ) ).
% insert_relcomp_fold
thf(fact_2769_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ( finite_finite @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C2 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G2 @ C2 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_2770_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_2771_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_2772_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C2: set @ ( set @ B ),G2: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ( finite_finite @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C2 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G2 @ C2 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_2773_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite @ A @ A1 )
=> ( ( finite_finite @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2774_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite @ A @ A1 )
& ( finite_finite @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A1 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_2775_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_2776_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R2: A > A > $o,X2: set @ A,Y2: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y2 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_2777_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: set @ A,Y2: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y2 ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_2778_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ A] :
( ( finite_finite @ ( 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 ) )
@ ^ [X2: A,Y2: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S ) @ ( insert2 @ B @ Y2 @ A8 ) @ A8 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_2779_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_2780_chains__extend,axiom,
! [A: $tType,C3: set @ ( set @ A ),S: set @ ( set @ A ),Z2: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S ) )
=> ( ( member @ ( set @ A ) @ Z2 @ S )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ X3 @ Z2 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) @ ( chains2 @ A @ S ) ) ) ) ) ).
% chains_extend
thf(fact_2781_ImageI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R3: set @ ( product_prod @ A @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R3 @ A3 ) ) ) ) ).
% ImageI
thf(fact_2782_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_2783_ex__assn__proper,axiom,
! [A: $tType,P: A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [X3: A] : ( proper @ ( P @ X3 ) )
=> ( proper
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X2: A] : ( P @ X2 @ H ) ) ) ).
% ex_assn_proper
thf(fact_2784_Image__empty1,axiom,
! [B: $tType,A: $tType,X6: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_2785_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B3: A,R3: set @ ( product_prod @ B @ A ),A4: B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R3 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B3 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_2786_ImageE,axiom,
! [A: $tType,B: $tType,B3: A,R3: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R3 @ A3 ) )
=> ~ ! [X3: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B3 ) @ R3 )
=> ~ ( member @ B @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_2787_Image__iff,axiom,
! [A: $tType,B: $tType,B3: A,R3: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R3 @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B3 ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_2788_rev__ImageI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R3 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_2789_Image__singleton,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A4: B] :
( ( image @ B @ A @ R3 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B4 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_2790_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R2: set @ ( product_prod @ A @ C ),S5: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Z3: B] :
? [Y2: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y2 ) @ R2 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z3 ) @ S5 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_2791_ex__assn__def,axiom,
! [A: $tType] :
( ( ex_assn @ A )
= ( ^ [P3: A > assn] :
( abs_assn
@ ^ [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X2: A] : ( rep_assn @ ( P3 @ X2 ) @ H ) ) ) ) ).
% ex_assn_def
thf(fact_2792_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ A ),B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y2: B] : ( image @ B @ A @ R2 @ ( insert2 @ B @ Y2 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_2793_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A,B4: A] :
( ( Uu
= ( inf_inf @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A3 )
& ( member @ A @ B4 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_2794_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A] :
( ( Uu
= ( inf_inf @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A3 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_2795_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A,B4: A] :
( ( Uu
= ( sup_sup @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A3 )
& ( member @ A @ B4 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_2796_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A] :
( ( Uu
= ( sup_sup @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A3 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_2797_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_2798_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R6: 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 )] :
? [X2: A,Y2: B] :
( ( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X2 ) @ ( product_Pair @ $o @ B @ $false @ Y2 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R6 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X2: A,Y2: B] :
( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X2 ) @ ( product_Pair @ $o @ B @ $false @ Y2 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_2799_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: 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 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_2800_subset__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_2801_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_2802_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_2803_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y3: $o] :
( ( member @ $o @ Y3 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P @ ( product_Abs_unit @ Y3 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_2804_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_2805_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_2806_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F2: B > A,X: B,C3: C,G2: B > C,A3: set @ B] :
( ( B3
= ( F2 @ X ) )
=> ( ( C3
= ( G2 @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_2807_subset__Image__Image__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ A3 ) @ ( image @ A @ A @ R3 @ B2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_2808_Rep__unit,axiom,
! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_2809_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y3: $o] :
( ( X
= ( product_Abs_unit @ Y3 ) )
=> ~ ( member @ $o @ Y3 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_2810_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ~ ! [X3: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).
% Rep_unit_cases
thf(fact_2811_in__lex__prod,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A7: A,B6: B,R3: set @ ( product_prod @ A @ A ),S2: 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 @ A4 @ B3 ) @ ( product_Pair @ A @ B @ A7 @ B6 ) ) @ ( lex_prod @ A @ B @ R3 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A7 ) @ R3 )
| ( ( A4 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2812_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_2813_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 ) ) ) )
| ? [A6: A] :
( R3
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_2814_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_2815_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A4 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_2816_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_2817_IdI,axiom,
! [A: $tType,A4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_2818_pair__in__Id__conv,axiom,
! [A: $tType,A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( id2 @ A ) )
= ( A4 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_2819_Linear__order__in__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2820_IdE,axiom,
! [A: $tType,P5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P5 @ ( id2 @ A ) )
=> ~ ! [X3: A] :
( P5
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_2821_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( id2 @ A ) )
=> ( A4 = B3 ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_2822_refl__onD,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( refl_on @ A @ A3 @ R3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_2823_refl__onD1,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A3 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_2824_refl__onD2,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A3 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_2825_refl__on__domain,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( refl_on @ A @ A3 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( member @ A @ A4 @ A3 )
& ( member @ A @ B3 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_2826_refl__on__empty,axiom,
! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% refl_on_empty
thf(fact_2827_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P7: product_prod @ A @ A] :
? [X2: A] :
( P7
= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_def
thf(fact_2828_partial__order__on__empty,axiom,
! [A: $tType] : ( order_7125193373082350890der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% partial_order_on_empty
thf(fact_2829_lnear__order__on__empty,axiom,
! [A: $tType] : ( order_679001287576687338der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% lnear_order_on_empty
thf(fact_2830_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: 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 ) @ Ra )
| ( ( A5 = A14 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_2831_reflcl__set__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_2832_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A4 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_2833_Zorns__po__lemma,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ ( chains @ A @ R3 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R3 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X4 ) @ R3 ) ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R3 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_2834_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 ) ) )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R3 ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2835_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B2: set @ A,F5: A > ( filter @ B ),P: B > $o] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ B2 )
=> ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X4 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A6 ) @ ( F5 @ B5 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ B2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( eventually @ B @ P @ ( F5 @ X2 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_2836_pair__lessI2,axiom,
! [A4: nat,B3: nat,S2: nat,T2: nat] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( ord_less @ nat @ S2 @ T2 )
=> ( 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 @ A4 @ S2 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_2837_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_2838_eventually__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] : P
@ F5 )
= P ) ) ).
% eventually_const
thf(fact_2839_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_2840_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R )
=> ( ! [F4: A > B,G4: A > B,X3: A,R4: B] :
( ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X3 ) @ R )
=> ( ( F4 @ Z6 )
= ( G4 @ Z6 ) ) )
=> ( ( P @ F4 @ X3 @ R4 )
= ( P @ G4 @ X3 @ R4 ) ) )
=> ( ! [X3: A,F4: A > B] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X_12: B] : ( P @ F4 @ X3 @ X_12 ) )
=> ? [F4: A > B] :
! [X4: A] : ( P @ F4 @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_2841_wf__induct__rule,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% wf_induct_rule
thf(fact_2842_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [Q3: set @ A] :
( ? [X2: A] : ( member @ A @ X2 @ Q3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ Q3 )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 )
=> ~ ( member @ A @ Y2 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2843_wf__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R3 ) ) ).
% wf_not_refl
thf(fact_2844_wf__not__sym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A4 ) @ R3 ) ) ) ).
% wf_not_sym
thf(fact_2845_wf__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R3 ) ) ).
% wf_irrefl
thf(fact_2846_wf__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% wf_induct
thf(fact_2847_wf__asym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A4 ) @ R3 ) ) ) ).
% wf_asym
thf(fact_2848_wfUNIVI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [P4: A > $o,X3: A] :
( ! [Xa2: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa2 ) @ R3 )
=> ( P4 @ Y3 ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf @ A @ R3 ) ) ).
% wfUNIVI
thf(fact_2849_wfI__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Q5: set @ A] :
( ( member @ A @ X3 @ Q5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Q5 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa2 ) @ R )
=> ~ ( member @ A @ Y3 @ Q5 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_min
thf(fact_2850_wfE__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
( ( wf @ A @ R )
=> ( ( member @ A @ X @ Q )
=> ~ ! [Z4: A] :
( ( member @ A @ Z4 @ Q )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z4 ) @ R )
=> ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_2851_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [P3: A > $o] :
( ! [X2: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 )
=> ( P3 @ Y2 ) )
=> ( P3 @ X2 ) )
=> ! [X7: A] : ( P3 @ X7 ) ) ) ) ).
% wf_def
thf(fact_2852_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_2853_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_2854_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_2855_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 )
=> ( X2 = Y2 ) ) ) ) ) ).
% antisym_def
thf(fact_2856_antisymI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
=> ( X3 = Y3 ) ) )
=> ( antisym @ A @ R3 ) ) ).
% antisymI
thf(fact_2857_antisymD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R3 )
=> ( A4 = B3 ) ) ) ) ).
% antisymD
thf(fact_2858_trivial__limit__def,axiom,
! [A: $tType,F5: filter @ A] :
( ( F5
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X2: A] : $false
@ F5 ) ) ).
% trivial_limit_def
thf(fact_2859_eventually__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : P
@ F5 )
= ( P
| ( F5
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_2860_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X3: A] :
~ ( P @ X3 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_2861_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_2862_wfE__min_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z4: A] :
( ( member @ A @ Z4 @ Q )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z4 ) @ R )
=> ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_2863_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_2864_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
~ ? [F: nat > A] :
! [I2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) @ R2 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_2865_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_2866_wf__bounded__measure,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R3 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ nat @ ( F2 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ nat @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R3 ) ) ).
% wf_bounded_measure
thf(fact_2867_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ ( image @ A @ A @ R @ A9 ) )
=> ( A9
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_2868_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ R @ A3 ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_2869_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_2870_eventually__Inf__base,axiom,
! [A: $tType,B2: set @ ( filter @ A ),P: A > $o] :
( ( B2
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B2 )
=> ! [G5: filter @ A] :
( ( member @ ( filter @ A ) @ G5 @ B2 )
=> ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B2 )
& ( ord_less_eq @ ( filter @ A ) @ X4 @ ( inf_inf @ ( filter @ A ) @ F6 @ G5 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B2 )
& ( eventually @ A @ P @ X2 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_2871_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [A8: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R2 ) )
& ( A8
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A8 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2872_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F2: A > ( set @ B )] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R3 )
=> ( ( finite_finite @ B @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ ( set @ B ) @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R3 ) ) ).
% wf_bounded_set
thf(fact_2873_pair__lessI1,axiom,
! [A4: nat,B3: nat,S2: nat,T2: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( 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 @ A4 @ S2 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_2874_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_2875_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P3: A > $o,R6: 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
@ ^ [X2: A,Y2: B] :
( ( X9 = X2 )
& ( P3 @ X2 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y2 ) @ ( R6 @ X2 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_2876_smin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( 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 ) ) @ XS @ 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 ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_2877_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 ) )
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) ) )
@ ( chains @ A @ R3 ) ) ) ).
% Chains_subset'
thf(fact_2878_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R2: 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 @ R2 ) )
& ( ( ( A12 = B1 )
& ( A23 = B22 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ B1 @ B22 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R2 @ B1 @ B22 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B1 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R2 @ B1 @ B22 ) )
& ( A12 = B1 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B22 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_2879_pair__leqI2,axiom,
! [A4: nat,B3: nat,S2: nat,T2: nat] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( ord_less_eq @ nat @ S2 @ T2 )
=> ( 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 @ A4 @ S2 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_2880_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y9: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ 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 @ Y9 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_2881_smin__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( X6
!= ( 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 ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_2882_pred__on_Ochain__empty,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o] : ( pred_chain @ A @ A3 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_2883_subset_Ochain__empty,axiom,
! [A: $tType,A3: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_2884_subset__Zorn__nonempty,axiom,
! [A: $tType,A15: set @ ( set @ A )] :
( ( A15
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C8: set @ ( set @ A )] :
( ( C8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A15 @ ( ord_less @ ( set @ A ) ) @ C8 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) @ A15 ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A15 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A15 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_2885_Union__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A15: set @ ( set @ A )] :
( ( finite_finite @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A15 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Union_in_chain
thf(fact_2886_Inter__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A15: set @ ( set @ A )] :
( ( finite_finite @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A15 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Inter_in_chain
thf(fact_2887_subset_Ochain__extend,axiom,
! [A: $tType,A3: set @ ( set @ A ),C2: set @ ( set @ A ),Z2: set @ A] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C2 )
=> ( ( member @ ( set @ A ) @ Z2 @ A3 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z5: set @ A] : Y4 = Z5
@ X3
@ Z2 ) )
=> ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_2888_pred__on_Ochain__extend,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o,C2: set @ A,Z2: A] :
( ( pred_chain @ A @ A3 @ P @ C2 )
=> ( ( member @ A @ Z2 @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ C2 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ X3
@ Z2 ) )
=> ( pred_chain @ A @ A3 @ P @ ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_2889_finite__subset__Union__chain,axiom,
! [A: $tType,A3: set @ A,B11: set @ ( set @ A ),A15: set @ ( set @ A )] :
( ( finite_finite @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A15 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ~ ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ B9 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_2890_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 ) )
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_2891_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 ) )
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) ) ) ) ).
% Chains_subset
thf(fact_2892_pair__leqI1,axiom,
! [A4: nat,B3: nat,S2: nat,T2: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( 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 @ A4 @ S2 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_2893_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( 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 ) ) @ XS @ 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 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_2894_wmin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( 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 ) ) @ XS @ 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 ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_2895_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y6: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X6: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y6 )
=> ( ( 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 ) ) @ X6 @ Y6 ) @ 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 @ X6 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_2896_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_2897_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_2898_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_2899_smax__emptyI,axiom,
! [Y6: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite @ ( product_prod @ nat @ nat ) @ Y6 )
=> ( ( Y6
!= ( 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 ) ) ) @ Y6 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_2900_wmin__emptyI,axiom,
! [X6: 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 ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_2901_wmax__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite @ ( product_prod @ nat @ nat ) @ X6 )
=> ( 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 ) ) ) @ X6 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_2902_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_2903_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
=> ( Phi @ A4 @ B3 ) )
=> ( ( ( A4 = B3 )
=> ( Phi @ A4 @ B3 ) )
=> ( Phi @ A4 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2904_product__fold,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite @ A @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: A,Z3: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y2: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) )
@ Z3
@ B2 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A3 ) ) ) ) ).
% product_fold
thf(fact_2905_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A8: set @ A,B7: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y2: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B7 @ X2 ) ) )
@ A8 ) ) ) ) ).
% Sigma_def
thf(fact_2906_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B2: C > ( set @ D ),F2: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( B2 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F2 @ ( B2 @ X2 ) )
@ A3 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A3 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( B2 @ X2 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2907_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B2: C > ( set @ D ),F2: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( B2 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F2 @ ( B2 @ X2 ) )
@ A3 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A3 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( B2 @ X2 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2908_SigmaI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ B @ B3 @ ( B2 @ A4 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_2909_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
= ( ( member @ A @ A4 @ A3 )
& ( member @ B @ B3 @ ( B2 @ A4 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_2910_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_2911_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_2912_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_2913_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F2: B > A,G2: B > A,A3: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( sup_sup @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ A3 ) )
= ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
& ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ A3 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_2914_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_2915_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_2916_fst__image__times,axiom,
! [B: $tType,A: $tType,B2: set @ B,A3: set @ A] :
( ( ( B2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
= A3 ) ) ) ).
% fst_image_times
thf(fact_2917_snd__image__times,axiom,
! [B: $tType,A: $tType,A3: set @ B,B2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_2918_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A3: set @ A,X6: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : X6 )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_2919_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,B2: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A4 @ A3 )
@ ^ [Uu: A] : ( insert2 @ B @ B3 @ B2 ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( insert2 @ B @ B3 @ B2 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A4 @ A3 )
@ ^ [Uu: A] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_2920_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B2 @ X3 ) )
=> ( C3
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_2921_SigmaD1,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ( member @ A @ A4 @ A3 ) ) ).
% SigmaD1
thf(fact_2922_SigmaD2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ( member @ B @ B3 @ ( B2 @ A4 ) ) ) ).
% SigmaD2
thf(fact_2923_SigmaE2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ~ ( ( member @ A @ A4 @ A3 )
=> ~ ( member @ B @ B3 @ ( B2 @ A4 ) ) ) ) ).
% SigmaE2
thf(fact_2924_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_2925_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B,C2: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 )
= ( product_Sigma @ A @ B @ C2
@ ^ [Uu: A] : D4 ) )
= ( ( ( A3 = C2 )
& ( B2 = D4 ) )
| ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C2
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_2926_wo__rel_OTOTALS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R3 ) )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa2 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X4 ) @ R3 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_2927_wo__rel_Omax2__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 )
= B3 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 )
= A4 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_2928_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_2929_well__order__induct__imp,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R3 ) )
=> ( ( member @ A @ Y5 @ ( field2 @ A @ R3 ) )
=> ( P @ Y5 ) ) )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( P @ X3 ) ) )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( P @ A4 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2930_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I4: set @ A,X6: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I4 @ X6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( ( X6 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_2931_wo__rel_Omax2__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 ) ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_2932_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 )
= B3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_2933_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 )
= A4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_2934_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( ord_less_eq @ A @ A4 @ ( complete_Inf_Inf @ A @ S ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ A4 @ X2 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_2935_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A3: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less_eq @ A @ X4 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_2936_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Y: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X6 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Y )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X6 )
& ( ord_less @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_2937_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A3: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less_eq @ A @ B5 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_2938_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S ) @ A4 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ord_less_eq @ A @ X2 @ A4 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_2939_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Y: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X6 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X6 )
& ( ord_less @ A @ Y @ X2 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_2940_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A3: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : A3 ) )
= ( finite_card @ B @ A3 ) ) ).
% card_cartesian_product_singleton
thf(fact_2941_times__subset__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,C2: set @ B,B2: set @ A,D4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C2 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : D4 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( C2
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
& ( ord_less_eq @ ( set @ B ) @ C2 @ D4 ) ) ) ) ).
% times_subset_iff
thf(fact_2942_wo__rel_Omax2__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_2943_wfI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : B2 ) )
=> ( ! [X3: A,P4: A > $o] :
( ! [Xa2: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa2 ) @ R3 )
=> ( P4 @ Y3 ) )
=> ( P4 @ Xa2 ) )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ X3 @ B2 )
=> ( P4 @ X3 ) ) ) )
=> ( wf @ A @ R3 ) ) ) ).
% wfI
thf(fact_2944_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite @ A @ A3 )
& ( finite_finite @ B @ B2 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_2945_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite @ B @ B2 ) ) ) ).
% finite_cartesian_productD2
thf(fact_2946_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite @ A @ A3 ) ) ) ).
% finite_cartesian_productD1
thf(fact_2947_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( B2 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( finite_finite @ B @ ( B2 @ A6 ) ) )
=> ( finite_finite @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% finite_SigmaI2
thf(fact_2948_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( B2 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_2949_refl__onI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) )
=> ( refl_on @ A @ A3 @ R3 ) ) ) ).
% refl_onI
thf(fact_2950_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R2 ) ) ) ) ) ).
% refl_on_def
thf(fact_2951_swap__product,axiom,
! [B: $tType,A: $tType,A3: set @ B,B2: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I2: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I2 ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B2 ) )
= ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : A3 ) ) ).
% swap_product
thf(fact_2952_wo__rel_Ocases__Total,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( Phi @ A4 @ B3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R3 )
=> ( Phi @ A4 @ B3 ) )
=> ( Phi @ A4 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2953_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_2954_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ B,F2: C > A,A3: set @ C,G2: B > A] :
( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F2 @ A3 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B2 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G2 @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_2955_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B2 ) @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_2956_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_2957_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G2: C > A,B2: set @ C,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G2 @ B2 ) )
=> ( ! [N4: B] :
( ( member @ B @ N4 @ A3 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( G2 @ X4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G2 @ B2 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_2958_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 ) ) @ R3 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A4 @ B3 ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_2959_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_2960_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A4: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X6 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ X6 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_2961_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A4: A] :
( ( condit1013018076250108175_below @ A @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ X6 ) )
= A4 ) )
& ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ X6 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_2962_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A4: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X6 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ X6 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_2963_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A4: A] :
( ( condit941137186595557371_above @ A @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ X6 ) )
= A4 ) )
& ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ X6 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_2964_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_2965_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_2966_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F2: B > A,A4: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ A4 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ ( F2 @ X2 ) @ A4 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_2967_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F2: B > A,A4: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ A4 @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_2968_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,G2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ A3 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_2969_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G2: D > B,A3: set @ C,B2: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X2: C,Y2: D] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G2 @ Y2 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu: C] : B2 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A3 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G2 @ B2 ) ) ) ).
% image_paired_Times
thf(fact_2970_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,G2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ A3 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A3 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_2971_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G2: B > A,B2: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G2 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ord_less_eq @ A @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_2972_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G2: B > A,B2: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G2 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G2 @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_2973_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_2974_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( inf_inf @ A @ ( F2 @ A4 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_2975_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_2976_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( sup_sup @ A @ ( F2 @ A4 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_2977_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ B2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2978_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ B2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2979_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R3 @ B2 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_2980_wo__rel_Ominim__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B2 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_2981_wo__rel_Ominim__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_2982_wo__rel_Oequals__minim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A4 @ B2 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B5 ) @ R3 ) )
=> ( A4
= ( bNF_We6954850376910717587_minim @ A @ R3 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_2983_wo__rel_Ominim__least,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B2 ) @ B3 ) @ R3 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_2984_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R3 @ B2 @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B2 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_2985_wo__rel_OisMinim__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R3 @ A3 @ B3 )
= ( ( member @ A @ B3 @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ X2 ) @ R3 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_2986_rtrancl__last__visit__node,axiom,
! [A: $tType,S2: A,S3: A,R: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ S3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ( S2 != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ S3 )
@ ( 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 @ S2 @ Sh ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S3 )
@ ( 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_2987_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > A,G2: C > B,A3: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ A3 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A3 )
@ ^ [X2: A] : ( image2 @ C @ B @ G2 @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_2988_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P3: A > A > $o,A8: 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 @ A8
@ ^ [Uu: A] : A8 ) )
& ( P3 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_2989_finite__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P7: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A5: A] :
( ( X2
= ( insert2 @ A @ A5 @ A8 ) )
& ( P7 @ A8 ) ) ) ) ) ).
% finite_def
thf(fact_2990_bsqr__max2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A1: A,A22: A,B13: A,B23: 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 @ B13 @ B23 ) ) @ ( 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 @ B13 @ B23 ) ) @ R3 ) ) ) ).
% bsqr_max2
thf(fact_2991_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_2992_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_2993_vimage__const,axiom,
! [B: $tType,A: $tType,C3: B,A3: set @ B] :
( ( ( member @ B @ C3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C3
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C3
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_2994_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E4: set @ ( product_prod @ B @ A )] :
( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E4 )
= ( image @ B @ A @ E4 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_2995_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_2996_vimage__if,axiom,
! [B: $tType,A: $tType,C3: B,A3: set @ B,D3: B,B2: set @ A] :
( ( ( member @ B @ C3 @ A3 )
=> ( ( ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D3 )
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D3 )
@ A3 )
= B2 ) ) ) )
& ( ~ ( member @ B @ C3 @ A3 )
=> ( ( ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D3 )
@ A3 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) ) )
& ( ~ ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B2 ) @ C3 @ D3 )
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_2997_well__order__on__domain,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_well_order_on @ A @ A3 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( member @ A @ A4 @ A3 )
& ( member @ A @ B3 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_2998_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_2999_converse__rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ B3 )
=> ( ! [Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ Z4 )
=> ( P @ Y3 ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_3000_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 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_3001_rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ A4 )
=> ( ! [Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( ( P @ Y3 )
=> ( P @ Z4 ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% rtrancl_induct
thf(fact_3002_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_3003_rtranclE,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A4 != B3 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B3 ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_3004_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_3005_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( transitive_rtrancl @ A @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_3006_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_3007_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 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B5 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_3008_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_3009_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A4: A,F2: A > B,B3: B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F2 @ A4 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_3010_well__order__on__empty,axiom,
! [A: $tType] : ( order_well_order_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% well_order_on_empty
thf(fact_3011_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q4: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( 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 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_3012_rtrancl__image__advance,axiom,
! [A: $tType,Q4: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ R )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_3013_rtrancl__Un__separatorE,axiom,
! [A: $tType,A4: A,B3: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Q )
=> ( X3 = Y3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_3014_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A4: A,B3: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B3 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ Q )
=> ( Y3 = X3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_3015_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 )
=> ( ! [A6: A,B5: 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 @ A6 @ B5 ) ) @ ( 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 @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R3 )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_3016_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za: 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 @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A6: A,B5: 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 @ A6 @ B5 ) ) @ 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 @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_3017_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 )
=> ( ! [A6: A,B5: 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 @ A6 @ B5 ) @ ( 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 @ A6 @ B5 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_3018_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A3: set @ B,F2: B > ( set @ A )] :
( ( ( member @ B @ X @ A3 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A3 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A3 @ F2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3019_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F2 @ A3 )
= ( bot_bot @ ( set @ B ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_3020_vimage__insert,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: B,B2: set @ B] :
( ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_3021_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_3022_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) )
=> ( ( A4 = B3 )
| ( member @ A @ A4 @ A3 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_3023_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),U3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( product_Sigma @ A @ A @ U3
@ ^ [Uu: A] : U3 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_3024_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite @ B @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F2 @ A3 ) )
& ~ ( finite_finite @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_3025_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite @ B @ A3 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( finite_finite @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_3026_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A3: set @ B] :
( ( finite_finite @ A @ F5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ F5 )
=> ( finite_finite @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) )
=> ( finite_finite @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A3 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_3027_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,E4: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F2 ) @ E4 ) ) ) ) ).
% rtrancl_mapI
thf(fact_3028_rtrancl__apply__insert,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,S: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( 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_3029_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F: A > B,B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y2: B] : ( vimage @ A @ B @ F @ ( insert2 @ B @ Y2 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3030_rtrancl__last__touch,axiom,
! [A: $tType,Q4: A,Q7: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 ) @ ( 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 @ Q7 )
@ ( 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_3031_rtrancl__last__visit_H,axiom,
! [A: $tType,Q4: A,Q7: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 )
@ ( 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 @ Q7 )
@ ( 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_3032_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite @ B @ A3 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( finite_finite @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3033_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite @ B @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F2 @ A3 ) )
& ~ ( finite_finite @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3034_rtrancl__insert,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A4 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ Y2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_3035_finite__reachable__advance,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),V0: A,V: A] :
( ( finite_finite @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( finite_finite @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_3036_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_3037_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 )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R3 ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3038_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( 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 @ E4 @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_3039_trancl__multi__insert2,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),M2: A,X6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : X6 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ M2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_3040_trancl__multi__insert,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,M2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ X6
@ ^ [Uu: A] : ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_3041_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_3042_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_3043_rel__restrict__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R @ ( bot_bot @ ( set @ A ) ) )
= R ) ).
% rel_restrict_empty
thf(fact_3044_trancl__empty,axiom,
! [A: $tType] :
( ( transitive_trancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trancl_empty
thf(fact_3045_trancl__single,axiom,
! [A: $tType,A4: A,B3: A] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_3046_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
=> ~ ( member @ A @ W2 @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_3047_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
=> ~ ( member @ A @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_3048_rel__restrict__trancl__mem,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A3 @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A3 ) @ R ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_3049_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 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B5 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_3050_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_3051_trancl_Or__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_3052_tranclE,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ~ ! [C4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C4 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C4 @ B3 ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_3053_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_3054_trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y3 ) @ R3 )
=> ( P @ Y3 ) )
=> ( ! [Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( ( P @ Y3 )
=> ( P @ Z4 ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% trancl_induct
thf(fact_3055_r__r__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_3056_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 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_3057_irrefl__trancl__rD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_3058_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_3059_trancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_3060_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 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( P @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ X3 @ Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ Y3 @ Z4 )
=> ( P @ X3 @ Z4 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_3061_converse__trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B3 ) @ R3 )
=> ( P @ Y3 ) )
=> ( ! [Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ Z4 )
=> ( P @ Y3 ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% converse_trancl_induct
thf(fact_3062_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 ) ) ).
% rel_restrict_lift
thf(fact_3063_rel__restrictI,axiom,
! [A: $tType,X: A,R: set @ A,Y: A,E4: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ R )
=> ( ~ ( member @ A @ Y @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_3064_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A3 @ R ) )
=> ~ ( member @ A @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_3065_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A3 @ R ) )
=> ~ ( member @ A @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_3066_less__eq,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% less_eq
thf(fact_3067_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E4 ) )
=> ( ~ ( member @ A @ X @ R )
=> ( ~ ( member @ A @ Y @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_3068_trancl__rtrancl__trancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_3069_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_3070_rtrancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_3071_rtrancl__into__trancl1,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_3072_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_3073_trancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_3074_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 ) )
=> ? [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z4 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Y ) @ R ) ) ) ).
% tranclD2
thf(fact_3075_rtranclD,axiom,
! [A: $tType,A4: A,B3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A4 = B3 )
| ( ( A4 != B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtranclD
thf(fact_3076_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 ) )
=> ? [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z4 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% tranclD
thf(fact_3077_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 ) )
=> ( ! [A6: A,B5: 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 @ A6 @ B5 ) ) @ R3 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: 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 @ A6 @ B5 ) ) @ ( 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 @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R3 )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_3078_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_3079_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M2: B > A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) )
=> ( ( P @ K )
=> ? [X3: B] :
( ( P @ X3 )
& ! [Y5: B] :
( ( P @ Y5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X3 ) @ ( M2 @ Y5 ) ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_3080_trancl__union__outside,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),U3: 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 ) ) @ E4 @ U3 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E4 ) )
=> ? [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ U3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U3 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_3081_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W2: A,V1: A,V22: A,E4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E4 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ E4 ) )
=> ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W2 ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_3082_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A8: 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 ) @ R6 )
& ~ ( member @ A @ V2 @ A8 )
& ~ ( member @ A @ W3 @ A8 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_3083_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),U3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( product_Sigma @ A @ A @ U3
@ ^ [Uu: A] : U3 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E4 ) ) ) ).
% Restr_trancl_mono
thf(fact_3084_rel__restrict__Int__empty,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ ( field2 @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R @ A3 )
= R ) ) ).
% rel_restrict_Int_empty
thf(fact_3085_rel__restrict__compl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A3 ) @ ( rel_restrict @ A @ R @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_3086_trancl__insert2,axiom,
! [A: $tType,A4: A,B3: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y2: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A4 ) @ ( transitive_trancl @ A @ R3 ) )
| ( X2 = A4 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ Y2 ) @ ( transitive_trancl @ A @ R3 ) )
| ( Y2 = B3 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_3087_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% uminus_rat_def
thf(fact_3088_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_3089_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ 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 @ E4 @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_3090_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ 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 ) ) @ E4
@ ( product_Sigma @ A @ A @ S
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_3091_rtrancl__last__visit,axiom,
! [A: $tType,Q4: A,Q7: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 )
@ ( 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 @ Q7 )
@ ( 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_3092_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_3093_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 )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y2 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) ) ) ) ).
% plus_rat_def
thf(fact_3094_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 )
@ ^ [X2: product_prod @ int @ int,Y2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y2 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y2 ) ) ) ) ) ).
% times_rat_def
thf(fact_3095_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [A5: A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_3096_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_3097_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F2
@ ^ [X2: A] : ( member @ A @ X2 @ S )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_3098_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A4: A,A3: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( insert2 @ A @ A4 @ A3 ) )
= ( F2 @ A4 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_3099_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A4: A,A3: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ A3 )
= ( F2 @ A4 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_3100_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R6: 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 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_3101_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_3102_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_3103_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_3104_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F: A > nat,R6: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
@ ^ [X2: A] : ( product_Pair @ nat @ A @ ( F @ X2 ) @ X2 ) ) ) ) ).
% mlex_prod_def
thf(fact_3105_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3106_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A3 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A3 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I4 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3107_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A3 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3108_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A3 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I4 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A3 @ I4 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3109_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_3110_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A4 ) ) ) ).
% mono_add
thf(fact_3111_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_3112_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_3113_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_3114_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% monoI
thf(fact_3115_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F: A > B] :
! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% mono_def
thf(fact_3116_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_min @ B @ ( F2 @ M2 ) @ ( F2 @ N ) )
= ( F2 @ ( ord_min @ A @ M2 @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_3117_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_3118_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F2: A > B,A3: A,B2: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A3 @ B2 ) ) @ ( inf_inf @ B @ ( F2 @ A3 ) @ ( F2 @ B2 ) ) ) ) ) ).
% mono_inf
thf(fact_3119_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F2: A > B,A3: A,B2: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A3 ) @ ( F2 @ B2 ) ) @ ( F2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% mono_sup
thf(fact_3120_Rings_Omono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A4 ) ) ) ) ).
% Rings.mono_mult
thf(fact_3121_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_3122_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_3123_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ B ),F: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) @ R2 ) ) ) ) ) ).
% inv_image_def
thf(fact_3124_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P7: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A5: A] :
( ( X2
= ( insert2 @ A @ A5 @ A8 ) )
& ( P7 @ A8 ) ) ) ) ).
% finite.mono
thf(fact_3125_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 ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A ),F: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F ) @ ( inv_image @ A @ B @ S7 @ F ) ) ) ) ).
% rp_inv_image_def
thf(fact_3126_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
@ ^ [X2: A] :
! [Y2: A] :
( ( member @ A @ Y2 @ S )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_3127_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
@ ^ [X2: A] :
! [Y2: A] :
( ( member @ A @ Y2 @ S )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_3128_flip__pred,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ A @ B ),R: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( 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 )
@ ^ [X2: A,Y2: B] : ( product_Pair @ B @ A @ Y2 @ X2 ) )
@ A3 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% flip_pred
thf(fact_3129_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_3130_ball__empty,axiom,
! [A: $tType,P: A > $o,X4: A] :
( ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X4 ) ) ).
% ball_empty
thf(fact_3131_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F2: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I2: A,X2: D] : ( Q @ I2 @ ( F2 @ X2 ) ) ) ) ) ).
% mono_compose
thf(fact_3132_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C9: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ C9 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ C9 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_3133_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y2: A,Z3: A] :
( ( member @ A @ Y2 @ A8 )
& ( member @ A @ Z3 @ A8 ) )
@ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R2 ) ) ) ) ) ).
% refl_on_def'
thf(fact_3134_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ( B4 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X2 ) @ R2 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_3135_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X2 ) @ R2 ) ) ) ) ) ) ).
% Under_def
thf(fact_3136_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P7: product_prod @ C @ D] : ( insert2 @ D @ ( product_snd @ C @ D @ P7 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_3137_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B4 ) @ R2 ) ) ) ) ) ) ).
% Above_def
thf(fact_3138_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X7: set @ A,Y10: set @ A] :
( ( Uu
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X7 @ Y10 ) )
& ( X7
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ Y10 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ X7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R2 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_3139_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X4: A] :
( ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X4 ) ) ).
% bex_empty
thf(fact_3140_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B ),S5: set @ A] :
( collect @ B
@ ^ [Y2: B] :
? [X2: A] :
( ( member @ A @ X2 @ S5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 ) ) ) ) ) ).
% Image_def
thf(fact_3141_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R @ A1 @ A22 )
=> ~ ( ( finite_finite @ A @ A1 )
=> ( ( finite_finite @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R @ X4 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_3142_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite @ A @ A12 )
& ( finite_finite @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A12 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A23 )
& ( R6 @ X2 @ Y2 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_3143_max__extp_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y6: set @ A,R: A > A > $o] :
( ( finite_finite @ A @ X6 )
=> ( ( finite_finite @ A @ Y6 )
=> ( ( Y6
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y6 )
& ( R @ X3 @ Xa2 ) ) )
=> ( max_extp @ A @ R @ X6 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_3144_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X7: set @ A,Y10: set @ A] :
( ( finite_finite @ A @ X7 )
& ( finite_finite @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_3145_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ( B4 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B4 ) @ R2 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_3146_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_3147_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_3148_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_3149_converse__iff,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( converse @ B @ A @ R3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ R3 ) ) ).
% converse_iff
thf(fact_3150_converse__empty,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_empty
thf(fact_3151_acyclic__empty,axiom,
! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% acyclic_empty
thf(fact_3152_converseI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ ( converse @ A @ B @ R3 ) ) ) ).
% converseI
thf(fact_3153_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 ) )
=> ~ ! [X3: B,Y3: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y3 @ X3 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R3 ) ) ) ).
% converseE
thf(fact_3154_converseD,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( converse @ B @ A @ R3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ R3 ) ) ).
% converseD
thf(fact_3155_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_3156_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_3157_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_3158_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_3159_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_3160_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_3161_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y2: A,X2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ).
% converse_unfold
thf(fact_3162_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: B,Y2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ ( converse @ A @ B @ R3 ) ) ) ) ).
% conversep_converse_eq
thf(fact_3163_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R3: set @ ( product_prod @ B @ B ),F2: B > A] :
( ! [A6: B,B5: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B5 ) @ R3 )
=> ( ord_less @ A @ ( F2 @ B5 ) @ ( F2 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R3 ) ) ) ).
% acyclicI_order
thf(fact_3164_cyclicE,axiom,
! [A: $tType,G2: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ G2 )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ G2 ) ) ) ).
% cyclicE
thf(fact_3165_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% acyclic_def
thf(fact_3166_acyclicI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( transitive_acyclic @ A @ R3 ) ) ).
% acyclicI
thf(fact_3167_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 ) ) ) ) ) ) ).
% converse_def
thf(fact_3168_AboveS__disjoint,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( order_AboveS @ A @ R3 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_3169_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_3170_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P7: product_prod @ A @ B] : ( insert2 @ A @ ( product_fst @ A @ B @ P7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_3171_wo__rel_Osuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R3 @ B2 )
!= B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B2 ) ) @ R3 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_3172_wo__rel_Osuc__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B2 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_3173_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B2 ) @ ( order_AboveS @ A @ R3 @ B2 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_3174_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ A,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A4 @ ( order_AboveS @ A @ R3 @ B2 ) )
=> ( ! [A16: A] :
( ( member @ A @ A16 @ ( order_AboveS @ A @ R3 @ B2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A16 ) @ R3 ) )
=> ( A4
= ( bNF_Wellorder_wo_suc @ A @ R3 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_3175_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_3176_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X2: A,Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R2 ) ) ) ) ) ).
% trans_def
thf(fact_3177_transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R3 ) ) )
=> ( trans @ A @ R3 ) ) ).
% transI
thf(fact_3178_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_3179_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_3180_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_3181_trans__singleton,axiom,
! [A: $tType,A4: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_3182_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y2: A,Y13: A] :
! [Z3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z3 @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y13 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Aa3 ) @ R2 ) )
@ Z3 ) )
@ X2 ) ) ) ) ).
% trans_join
thf(fact_3183_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A4 @ ( order_AboveS @ A @ R3 @ B2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B2 ) @ A4 ) @ R3 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_3184_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),A3: set @ C,B2: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R3 @ ( B2 @ X2 ) )
@ A3 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_3185_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A3 )
=> ( ( ( order_AboveS @ A @ R3 @ A3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ A3 ) ) @ R3 )
=> ( ( B3
!= ( bNF_Wellorder_wo_suc @ A @ R3 @ A3 ) )
=> ( member @ A @ B3 @ A3 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_3186_wf__finite__segments,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R3 )
=> ( ( trans @ A @ R3 )
=> ( ! [X3: A] :
( finite_finite @ A
@ ( collect @ A
@ ^ [Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R3 ) ) )
=> ( wf @ A @ R3 ) ) ) ) ).
% wf_finite_segments
thf(fact_3187_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
@ ^ [R2: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S5 )
& ! [A5: A,B4: A,C5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ S5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R2 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_3188_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_3189_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
! [X2: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 )
=> ! [Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) ) ) ) ).
% single_valued_def
thf(fact_3190_single__valuedI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z4 ) @ R3 )
=> ( Y3 = Z4 ) ) )
=> ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedI
thf(fact_3191_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_3192_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R2 ) ) ) ).
% irrefl_def
thf(fact_3193_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_3194_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_3195_trans__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) @ ( 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 ) ) @ S2 @ T2 ) @ ( 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 @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_3196_antisym__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: 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 @ S2 ) @ ( 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 ) ) @ S2 @ R3 ) @ ( init_seg_of @ A ) )
=> ( R3 = S2 ) ) ) ).
% antisym_init_seg_of
thf(fact_3197_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_3198_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_3199_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_3200_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_3201_initial__segment__of__Diff,axiom,
! [A: $tType,P5: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ A @ A ),S2: 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 ) ) @ P5 @ 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 ) ) @ P5 @ S2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q4 @ S2 ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_3202_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K4: nat,M: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M @ K4 ) @ ( product_Pair @ nat @ nat @ M @ ( minus_minus @ nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus @ nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_3203_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_3204_ofilter__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A3 )
=> ( ( ord_less @ ( set @ 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 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_3205_ofilter__subset__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A3 )
=> ( ( order_ofilter @ A @ R3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
= ( 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 @ A3
@ ^ [Uu: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu: A] : B2 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_3206_ofilter__subset__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A3 )
=> ( ( order_ofilter @ A @ R3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
= ( 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 @ A3
@ ^ [Uu: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_3207_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 ) )
= ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_3208_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3209_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_3210_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_3211_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_3212_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_3213_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_3214_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_3215_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_3216_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_3217_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_3218_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_3219_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : ( single_valuedp @ A @ B @ ( bot_bot @ ( A > B > $o ) ) ) ).
% single_valuedp_bot
thf(fact_3220_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_3221_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_3222_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_3223_exists__minim__Well__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
=> ( order_well_order_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
& ! [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 ) ) @ X3 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_3224_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_finite @ A @ ( field2 @ A @ R3 ) )
=> ( ~ ( finite_finite @ 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_3225_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 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( 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 @ X2 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ X2 ) ) )
@ R5 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_3226_underS__Restr__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: 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 @ A4 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ A4 ) ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_3227_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R2 ) ) ) ) ) ).
% aboveS_def
thf(fact_3228_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A8: set @ A,R6: 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 @ A8 )
& ( member @ A @ A23 @ A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A12 ) @ ( F @ A23 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_3229_normalize__stable,axiom,
! [Q4: int,P5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
=> ( ( algebr8660921524188924756oprime @ int @ P5 @ Q4 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
= ( product_Pair @ int @ int @ P5 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_3230_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( ( algebr8660921524188924756oprime @ A @ A4 @ C3 )
& ( algebr8660921524188924756oprime @ A @ B3 @ C3 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_3231_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C3 @ ( times_times @ A @ A4 @ B3 ) )
= ( ( algebr8660921524188924756oprime @ A @ C3 @ A4 )
& ( algebr8660921524188924756oprime @ A @ C3 @ B3 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_3232_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ A4 )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_3233_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( gcd_gcd @ A @ A4 @ B3 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_3234_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A4 )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_3235_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_3236_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_3237_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_3238_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% is_unit_gcd
thf(fact_3239_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A4 ) ) ).
% coprime_1_left
thf(fact_3240_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ A4 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_3241_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_3242_underS__E,axiom,
! [A: $tType,I: A,R: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I @ ( order_underS @ A @ R @ J ) )
=> ( ( I != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) ) ) ).
% underS_E
thf(fact_3243_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R2 ) ) ) ) ) ).
% underS_def
thf(fact_3244_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ A4 @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_3245_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ A4 ) ) ).
% coprime_add_one_left
thf(fact_3246_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ A4 @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_3247_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ A4 ) ) ).
% coprime_diff_one_left
thf(fact_3248_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C3 )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_3249_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C3 )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_3250_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C3 )
=> ( ( dvd_dvd @ A @ B3 @ C3 )
=> ( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ) ) ).
% divides_mult
thf(fact_3251_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_3252_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_3253_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C3 @ A4 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_3254_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y: A,X: A] :
( ( dvd_dvd @ A @ Y @ X )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_3255_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,D3: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A4 )
=> ( ( dvd_dvd @ A @ E2 @ B3 )
=> ( dvd_dvd @ A @ E2 @ C3 ) ) ) )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A4 )
=> ( ( dvd_dvd @ A @ E2 @ B3 )
=> ( dvd_dvd @ A @ E2 @ D3 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_3256_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_3257_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ A4 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ~ ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% not_coprimeI
thf(fact_3258_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ~ ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A4 )
=> ( ( dvd_dvd @ A @ C4 @ B3 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_3259_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_3260_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A4 )
=> ( ( dvd_dvd @ A @ C4 @ B3 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% coprimeI
thf(fact_3261_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A4: A,N: A,M2: A,B3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ N ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ B3 @ N ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A4 @ M2 )
= ( modulo_modulo @ A @ B3 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_3262_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A4: A,M2: A,B3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A4 ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B3 ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A4 @ M2 )
= ( modulo_modulo @ A @ B3 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_3263_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C3 )
=> ( ( gcd_gcd @ A @ A4 @ ( times_times @ A @ B3 @ C3 ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_3264_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C3 )
=> ( ( gcd_gcd @ A @ A4 @ ( times_times @ A @ C3 @ B3 ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_3265_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( algebr8660921524188924756oprime @ A @ B3 @ C3 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C3 ) @ B3 )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_3266_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( algebr8660921524188924756oprime @ A @ B3 @ C3 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A4 ) @ B3 )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_3267_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_3268_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_3269_quotient__of__coprime,axiom,
! [R3: rat,P5: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P5 @ Q4 ) ) ).
% quotient_of_coprime
thf(fact_3270_normalize__coprime,axiom,
! [R3: product_prod @ int @ int,P5: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P5 @ Q4 ) ) ).
% normalize_coprime
thf(fact_3271_underS__empty,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( order_underS @ A @ R3 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_3272_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ C3 ) ) ) ).
% invertible_coprime
thf(fact_3273_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ? [A16: A,B8: A] :
( ( A4
= ( times_times @ A @ A16 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
& ( B3
= ( times_times @ A @ B8 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
& ( algebr8660921524188924756oprime @ A @ A16 @ B8 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_3274_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,A7: A,B6: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A4
= ( times_times @ A @ A7 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
=> ( ( B3
= ( times_times @ A @ B6 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A7 @ B6 ) ) ) ) ) ).
% gcd_coprime
thf(fact_3275_underS__Field3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R3 @ A4 ) @ ( field2 @ A @ R3 ) ) ) ).
% underS_Field3
thf(fact_3276_underS__incl__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A4 ) @ ( order_underS @ A @ R3 @ B3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_3277_underS__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( trans @ A @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A4 ) @ ( order_underS @ A @ R3 @ B3 ) ) ) ) ) ).
% underS_incr
thf(fact_3278_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 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ ( 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 @ X2 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ X2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_3279_Refl__under__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ( ( order_under @ A @ R3 @ A4 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R3 @ A4 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_3280_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,B2: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F2 @ B2 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_3281_antisymp__antisym__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( antisym @ A @ R3 ) ) ).
% antisymp_antisym_eq
thf(fact_3282_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_3283_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A4 @ B3 ) @ B3 )
= ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.right_idem
thf(fact_3284_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_max @ A @ A4 @ ( ord_max @ A @ A4 @ B3 ) )
= ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.left_idem
thf(fact_3285_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A] :
( ( ord_max @ A @ A4 @ A4 )
= A4 ) ) ).
% max.idem
thf(fact_3286_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A4 )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% max.bounded_iff
thf(fact_3287_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_max @ A @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb2
thf(fact_3288_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_max @ A @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb1
thf(fact_3289_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_3290_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_max @ A @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb4
thf(fact_3291_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_max @ A @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb3
thf(fact_3292_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_3293_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_3294_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_3295_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_3296_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_3297_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_3298_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_3299_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_3300_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_3301_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_3302_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_3303_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_3304_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max_insert
thf(fact_3305_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_max @ A @ ( ord_min @ A @ B3 @ C3 ) @ A4 )
= ( ord_min @ A @ ( ord_max @ A @ B3 @ A4 ) @ ( ord_max @ A @ C3 @ A4 ) ) ) ) ).
% max_min_distrib1
thf(fact_3306_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_max @ A @ A4 @ ( ord_min @ A @ B3 @ C3 ) )
= ( ord_min @ A @ ( ord_max @ A @ A4 @ B3 ) @ ( ord_max @ A @ A4 @ C3 ) ) ) ) ).
% max_min_distrib2
thf(fact_3307_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_min @ A @ ( ord_max @ A @ B3 @ C3 ) @ A4 )
= ( ord_max @ A @ ( ord_min @ A @ B3 @ A4 ) @ ( ord_min @ A @ C3 @ A4 ) ) ) ) ).
% min_max_distrib1
thf(fact_3308_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_min @ A @ A4 @ ( ord_max @ A @ B3 @ C3 ) )
= ( ord_max @ A @ ( ord_min @ A @ A4 @ B3 ) @ ( ord_min @ A @ A4 @ C3 ) ) ) ) ).
% min_max_distrib2
thf(fact_3309_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_3310_complete__linorder__sup__max,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% complete_linorder_sup_max
thf(fact_3311_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_3312_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_max @ A @ B3 @ ( ord_max @ A @ A4 @ C3 ) )
= ( ord_max @ A @ A4 @ ( ord_max @ A @ B3 @ C3 ) ) ) ) ).
% max.left_commute
thf(fact_3313_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( ord_max @ A @ B4 @ A5 ) ) ) ) ).
% max.commute
thf(fact_3314_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A4 @ B3 ) @ C3 )
= ( ord_max @ A @ A4 @ ( ord_max @ A @ B3 @ C3 ) ) ) ) ).
% max.assoc
thf(fact_3315_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_3316_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less @ A @ ( ord_max @ A @ B3 @ C3 ) @ A4 )
=> ~ ( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ C3 @ A4 ) ) ) ) ).
% max.strict_boundedE
thf(fact_3317_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_3318_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less @ A @ C3 @ A4 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_3319_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_3320_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_3321_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_3322_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A4: A,D3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A4 )
=> ( ( ord_less_eq @ A @ D3 @ B3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D3 ) @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ) ).
% max.mono
thf(fact_3323_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4
= ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.orderE
thf(fact_3324_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( ord_max @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% max.orderI
thf(fact_3325_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A4: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A4 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A4 )
=> ~ ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% max.boundedE
thf(fact_3326_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C3 @ A4 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A4 ) ) ) ) ).
% max.boundedI
thf(fact_3327_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_3328_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.cobounded1
thf(fact_3329_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.cobounded2
thf(fact_3330_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_3331_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_3332_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_3333_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A4 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.coboundedI1
thf(fact_3334_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.coboundedI2
thf(fact_3335_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_3336_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_3337_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_3338_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_3339_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_3340_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_max @ B @ ( F2 @ M2 ) @ ( F2 @ N ) )
= ( F2 @ ( ord_max @ A @ M2 @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_3341_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.abel_semigroup_axioms
thf(fact_3342_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_max @ A ) ) ) ).
% max.semilattice_axioms
thf(fact_3343_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.semigroup_axioms
thf(fact_3344_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R2 ) ) ) ) ).
% under_def
thf(fact_3345_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_3346_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_3347_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_3348_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_3349_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_3350_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_3351_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_3352_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_3353_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),R7: 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 @ R7 ) @ ( 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 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_3354_Sup__fin__def,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% Sup_fin_def
thf(fact_3355_antisym__bot,axiom,
! [A: $tType] : ( antisymp @ A @ ( bot_bot @ ( A > A > $o ) ) ) ).
% antisym_bot
thf(fact_3356_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_3357_finite__well__order__on__ordIso,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ A @ A3 )
=> ( ( order_well_order_on @ A @ A3 @ R3 )
=> ( ( order_well_order_on @ A @ A3 @ 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_3358_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_3359_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_3360_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P5 )
= ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P5 )
= ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_3361_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P5 )
= ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P5 )
= ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_3362_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_3363_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_3364_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite @ 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_3365_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A3 )
=> ( Less_eq @ X @ A10 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_3366_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( Less_eq @ X @ A6 ) )
=> ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_3367_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_3368_under__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R3 @ A4 ) @ ( order_under @ A @ R3 @ B3 ) ) ) ) ).
% under_incr
thf(fact_3369_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N3: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N3 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H2 @ N3 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_3370_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B2 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_3371_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_3372_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y3 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Max.closed
thf(fact_3373_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_3374_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 ) )
= ( ? [P7: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P7 ) @ ( 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 @ P7 ) @ ( 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 ) ) @ P7 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_3375_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 ) )
= ( ? [P7: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P7 ) @ ( 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 @ P7 ) @ ( 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 ) ) @ P7 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_3376_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_3377_semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A3 )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_3378_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_3379_semilattice__set_Ounion,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,B2: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ B2 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_3380_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_3381_semilattice__set_Oinsert,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_3382_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% Sup_fin.semilattice_set_axioms
thf(fact_3383_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_3384_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F2: A > A > A,H2: A > A,N3: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ! [X3: A,Y3: A] :
( ( H2 @ ( F2 @ X3 @ Y3 ) )
= ( F2 @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ A @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ N3 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ ( image2 @ A @ A @ H2 @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_3385_semilattice__set_Osubset,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,B2: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ B2 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_3386_semilattice__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( F2 @ X3 @ Y3 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_3387_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > A] :
( ( bind2 @ B @ A @ A3
@ ^ [X2: B] : ( insert2 @ A @ ( F2 @ X2 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F2 @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_3388_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3389_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_3390_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_3391_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_3392_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_3393_empty__bind,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A )] :
( ( bind2 @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_3394_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_3395_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_3396_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_3397_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_3398_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int,B3: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B3 ) )
= B3 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_3399_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_3400_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num,B3: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B3 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B3 ) ) ) ).
% power_int_add_numeral2
thf(fact_3401_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_3402_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_3403_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_one_over
thf(fact_3404_bind__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,B2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind2 @ B @ A @ A3
@ ^ [Uu: B] : B2 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind2 @ B @ A @ A3
@ ^ [Uu: B] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_3405_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind2 @ A @ B @ A3
@ ^ [Uu: A] : B2 )
= B2 ) ) ).
% nonempty_bind_const
thf(fact_3406_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( ( M2
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( one_one @ A ) ) )
& ( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_3407_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N3: int,A4: A] :
( ( ord_less_eq @ int @ N @ N3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A4 @ N ) @ ( power_int @ A @ A4 @ N3 ) ) ) ) ) ).
% power_int_increasing
thf(fact_3408_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N3: int,A4: A] :
( ( ord_less @ int @ N @ N3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N ) @ ( power_int @ A @ A4 @ N3 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_3409_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_3410_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: int,B3: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A4 @ B3 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B3 @ A4 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_3411_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N3: int,A4: A] :
( ( ord_less @ int @ N @ N3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N3 ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_3412_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_3413_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_3414_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_3415_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A4: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A4 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_3416_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_3417_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N3: int,A4: A] :
( ( ord_less_eq @ int @ N @ N3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( A4
!= ( zero_zero @ A ) )
| ( N3
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A4 @ N3 ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_3418_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_3419_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_3420_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_3421_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( inj_on @ A @ B @ F2 @ A3 )
& ( inj_on @ A @ B @ F2 @ B2 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_3422_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A8 ) ) ) ) ).
% quotient_def
thf(fact_3423_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ 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_3424_mset__set__Union,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( finite_finite @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A3 ) @ ( mset_set @ A @ B2 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_3425_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_3426_mset__set_Oempty,axiom,
! [A: $tType] :
( ( mset_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset_set.empty
thf(fact_3427_quotient__is__empty2,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A3 @ R3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_3428_quotient__is__empty,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A3 @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_3429_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_3430_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A4 ) @ ( top_top @ ( set @ A ) ) )
= ( A4
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_3431_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ A4 @ A3 ) )
= ( ( inj_on @ A @ B @ F2 @ A3 )
& ~ ( member @ B @ ( F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_3432_quotient__diff1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A5: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ R3 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A3 @ R3 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) ) ) ) ) ).
% quotient_diff1
thf(fact_3433_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( C3 @ X2 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_3434_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X2: A] : ( product_Pair @ B @ A @ ( C3 @ X2 ) @ X2 )
@ S ) ).
% inj_Pair(2)
thf(fact_3435_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X6: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( F2 @ X2 ) )
@ X6 ) ).
% inj_on_convol_ident
thf(fact_3436_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A,A3: set @ A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A4 ) @ A3 ) ) ) ).
% inj_on_mult
thf(fact_3437_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_3438_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ A ),F2: A > B] :
( ( S
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A9: set @ A] :
( ( member @ ( set @ A ) @ A9 @ S )
=> ( inj_on @ A @ B @ F2 @ A9 ) )
=> ( inj_on @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_3439_inj__singleton,axiom,
! [A: $tType,A3: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X2: A] : ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) ).
% inj_singleton
thf(fact_3440_quotient__of__eq,axiom,
! [A4: int,B3: int,P5: int,Q4: int] :
( ( ( quotient_of @ ( fract @ A4 @ B3 ) )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( ( fract @ P5 @ Q4 )
= ( fract @ A4 @ B3 ) ) ) ).
% quotient_of_eq
thf(fact_3441_quotientE,axiom,
! [A: $tType,X6: set @ A,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ~ ! [X3: A] :
( ( X6
= ( image @ A @ A @ R3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X3 @ A3 ) ) ) ).
% quotientE
thf(fact_3442_quotientI,axiom,
! [A: $tType,X: A,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A3 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A3 @ R3 ) ) ) ).
% quotientI
thf(fact_3443_swap__inj__on,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I2: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I2 ) )
@ A3 ) ).
% swap_inj_on
thf(fact_3444_normalize__eq,axiom,
! [A4: int,B3: int,P5: int,Q4: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A4 @ B3 ) )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( ( fract @ P5 @ Q4 )
= ( fract @ A4 @ B3 ) ) ) ).
% normalize_eq
thf(fact_3445_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_3446_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,A17: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F: A > B] :
( ( inj_on @ A @ B @ F @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A3 ) @ A17 ) ) )
= ( ? [G: B > A] :
( ( image2 @ B @ A @ G @ A17 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3447_card__quotient__disjoint,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ A @ A3 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A3 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A3 @ R3 ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_3448_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P5: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P5 ) @ P5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P5 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_3449_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F2: A > A,P5: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ P5 @ ( F2 @ P5 ) )
=> ( ord_less_eq @ A @ P5 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_3450_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_3451_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,F2: B > C,A3: A > ( set @ B )] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ( inj_on @ B @ C @ F2 @ ( A3 @ I3 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_3452_mset__set__empty__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( mset_set @ A @ A3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite @ A @ A3 ) ) ) ).
% mset_set_empty_iff
thf(fact_3453_quotient__of__Fract,axiom,
! [A4: int,B3: int] :
( ( quotient_of @ ( fract @ A4 @ B3 ) )
= ( normalize @ ( product_Pair @ int @ int @ A4 @ B3 ) ) ) ).
% quotient_of_Fract
thf(fact_3454_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_3455_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A4: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A4 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A4 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_3456_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_3457_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M2: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M2 @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_3458_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_3459_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,G2: A > B,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( inj_on @ A @ B @ G2 @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A3 ) @ ( image2 @ A @ B @ G2 @ B2 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A3 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_3460_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X2: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X2
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa4 @ X2 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_3461_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_3462_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,A4: B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) @ A3 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( F2 @ X2 )
= A4 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_3463_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= A4 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_3464_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_3465_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_3466_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ A ),X2: B] : ( image @ B @ A @ R2 @ ( insert2 @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_3467_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_3468_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F: A > B] :
! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_3469_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_3470_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_3471_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_3472_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R2: 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 ) @ R2 ) ) ) ) ) ).
% dir_image_def
thf(fact_3473_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_3474_lfp__induct2,axiom,
! [A: $tType,B: $tType,A4: A,B3: 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 @ A4 @ B3 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F2 ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F2 )
=> ( ! [A6: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( 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 @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% lfp_induct2
thf(fact_3475_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_3476_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_3477_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X7: set @ A] :
( the @ A
@ ^ [X2: A] :
( X7
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3478_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_3479_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( Less_eq2 @ X2 @ Y2 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_3480_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,G2: C > B,A3: set @ A,B2: 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 @ A3
@ ^ [Uu: A] : B2 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ B2 ) )
= ( 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 @ A3 )
& ( member @ C @ B4 @ B2 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_3481_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B4: A,A8: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A8 @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4
@ ( the @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_3482_proj__iff,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 )
=> ( ( ( 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_3483_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_3484_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_3485_in__quotient__imp__closed,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ A @ X @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ X6 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_3486_quotient__eq__iff,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,Y6: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ A @ X @ X6 )
=> ( ( member @ A @ Y @ Y6 )
=> ( ( X6 = Y6 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_3487_quotient__eqI,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,Y6: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ A @ X @ X6 )
=> ( ( member @ A @ Y @ Y6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X6 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_3488_in__quotient__imp__non__empty,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( X6
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_3489_equiv__class__self,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ A @ A4 @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_3490_quotient__disj,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,Y6: set @ A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( X6 = Y6 )
| ( ( inf_inf @ ( set @ A ) @ X6 @ Y6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_3491_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_3492_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_3493_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_3494_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_3495_eq__equiv__class,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A,A3: set @ A] :
( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ A @ B3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_3496_equiv__class__eq,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_3497_eq__equiv__class__iff,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( 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_3498_equiv__class__eq__iff,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ 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 @ A3 )
& ( member @ A @ Y @ A3 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_3499_eq__equiv__class__iff2,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( 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_3500_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A3 @ R )
=> ( ( equiv_equiv @ A @ A3 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_3501_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A3 @ R )
=> ( ( equiv_equiv @ A @ A3 @ S )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_3502_equiv__class__subset,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_3503_subset__equiv__class,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),B3: A,A4: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_3504_equiv__class__nondisjoint,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_3505_in__quotient__imp__in__rel,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_3506_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A18: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A18 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F2 )
=> ( ( member @ A @ A1 @ A18 )
=> ( ( member @ B @ A22 @ A25 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X13 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert2 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_3507_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),A4: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A4 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_3508_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),X6: set @ A,Y6: set @ A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F2 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X6 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ Y6 ) ) )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) )
=> ( X6 = Y6 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_3509_disjnt__equiv__class,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_3510_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A18: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A4: B] :
( ( equiv_equiv @ A @ A18 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F2 )
=> ( ( member @ B @ A4 @ A25 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X13 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_3511_disjnt__self__iff__empty,axiom,
! [A: $tType,S: set @ A] :
( ( disjnt @ A @ S @ S )
= ( S
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjnt_self_iff_empty
thf(fact_3512_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C2: set @ A,A3: set @ B,B2: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C2
@ ^ [Uu: A] : A3 )
@ ( product_Sigma @ A @ B @ C2
@ ^ [Uu: A] : B2 ) )
= ( ( C2
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A3 @ B2 ) ) ) ).
% disjnt_Times1_iff
thf(fact_3513_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,C2: set @ B,B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C2 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : C2 ) )
= ( ( C2
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A3 @ B2 ) ) ) ).
% disjnt_Times2_iff
thf(fact_3514_disjnt__empty1,axiom,
! [A: $tType,A3: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% disjnt_empty1
thf(fact_3515_disjnt__empty2,axiom,
! [A: $tType,A3: set @ A] : ( disjnt @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_3516_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,C2: A > ( set @ B ),B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ C2 ) @ ( product_Sigma @ A @ B @ B2 @ C2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( ( C2 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A3 @ B2 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_3517_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( inf_inf @ ( set @ A ) @ A8 @ B7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_3518_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_3519_congruentI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F2: A > B] :
( ! [Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( ( F2 @ Y3 )
= ( F2 @ Z4 ) ) )
=> ( equiv_congruent @ A @ B @ R3 @ F2 ) ) ).
% congruentI
thf(fact_3520_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F2: A > B > C,Y1: A,Z1: A,Y22: B,Z22: B] :
( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z22 ) @ R22 )
=> ( ( F2 @ Y1 @ Y22 )
= ( F2 @ Z1 @ Z22 ) ) ) ) ) ).
% congruent2D
thf(fact_3521_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: 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 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z23 ) @ R22 )
=> ( ( F2 @ Y12 @ Y23 )
= ( F2 @ Z12 @ Z23 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F2 ) ) ).
% congruent2I'
thf(fact_3522_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > A > B] :
( ( equiv_equiv @ A @ A3 @ R3 )
=> ( ! [Y3: A,Z4: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ( member @ A @ Z4 @ A3 )
=> ( ( F2 @ Y3 @ Z4 )
= ( F2 @ Z4 @ Y3 ) ) ) )
=> ( ! [Y3: A,Z4: A,W: A] :
( ( member @ A @ W @ A3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( ( F2 @ W @ Y3 )
= ( F2 @ W @ Z4 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R3 @ R3 @ F2 ) ) ) ) ).
% congruent2_commuteI
thf(fact_3523_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A18: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > C] :
( ( equiv_equiv @ A @ A18 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ! [Y3: A,Z4: A,W: B] :
( ( member @ B @ W @ A25 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R1 )
=> ( ( F2 @ Y3 @ W )
= ( F2 @ Z4 @ W ) ) ) )
=> ( ! [Y3: B,Z4: B,W: A] :
( ( member @ A @ W @ A18 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z4 ) @ R22 )
=> ( ( F2 @ W @ Y3 )
= ( F2 @ W @ Z4 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F2 ) ) ) ) ) ).
% congruent2I
thf(fact_3524_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A18: set @ B,B14: set @ A,F22: C > D,B24: set @ C,A25: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A18 )
= B14 )
=> ( ( inj_on @ C @ D @ F22 @ B24 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B24 ) @ A25 )
=> ( ( ( B24
= ( bot_bot @ ( set @ C ) ) )
=> ( A25
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B24 @ B14 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B24 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A25 @ A18 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3525_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G2: A > B,C2: set @ A,B2: set @ A,X: A] :
( ( inj_on @ A @ B @ G2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I2: B] : ( if @ A @ ( member @ B @ I2 @ ( image2 @ A @ B @ G2 @ C2 ) ) @ ( the_inv_into @ A @ B @ C2 @ G2 @ I2 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3526_mset__set_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( mset_set @ A @ ( insert2 @ A @ X @ A3 ) )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% mset_set.insert_remove
thf(fact_3527_mset__set_Oremove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( mset_set @ A @ A3 )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_3528_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_3529_Func__non__emp,axiom,
! [A: $tType,B: $tType,B2: set @ A,A3: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_3530_Func__is__emp,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A3 @ B2 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_3531_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,X6: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ X6 @ ( F2 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F2 ) ) ) )
=> ( ord_less_eq @ A @ X6 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ).
% coinduct
thf(fact_3532_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,F2: A > A,X6: A] :
( ( A3
= ( complete_lattice_gfp @ A @ F2 ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ X6 @ ( F2 @ ( sup_sup @ A @ X6 @ A3 ) ) )
=> ( ord_less_eq @ A @ X6 @ A3 ) ) ) ) ) ).
% def_coinduct
thf(fact_3533_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X6: A,F2: A > A] :
( ( ord_less_eq @ A @ X6 @ ( F2 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F2 ) ) ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F2 ) ) @ ( F2 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_3534_su__rel__fun_Of__def,axiom,
! [A: $tType,B: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( F2 @ A3 )
= ( the @ B
@ ^ [B7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ F5 ) ) ) ) ).
% su_rel_fun.f_def
thf(fact_3535_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B] :
( ! [A9: A,B9: B,B15: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B15 ) @ F5 )
=> ( B9 = B15 ) ) )
=> ( ! [A9: A,P4: $o] :
( ! [B16: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B16 ) @ F5 )
=> P4 )
=> P4 )
=> ( ! [A9: A] :
( ( F2 @ A9 )
= ( the @ B
@ ^ [B7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B7 ) @ F5 ) ) )
=> ( su_rel_fun @ A @ B @ F5 @ F2 ) ) ) ) ).
% su_rel_fun.intro
thf(fact_3536_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A,B2: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( ( F2 @ A3 )
= B2 )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ F5 ) ) ) ).
% su_rel_fun.repr
thf(fact_3537_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ ( F2 @ A3 ) ) @ F5 ) ) ).
% su_rel_fun.repr1
thf(fact_3538_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A,B2: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ F5 )
=> ( B2
= ( F2 @ A3 ) ) ) ) ).
% su_rel_fun.repr2
thf(fact_3539_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A,B2: B,B17: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ F5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B17 ) @ F5 )
=> ( B2 = B17 ) ) ) ) ).
% su_rel_fun.unique
thf(fact_3540_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ~ ! [B9: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B9 ) @ F5 ) ) ).
% su_rel_fun.surjective
thf(fact_3541_su__rel__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( su_rel_fun @ A @ B )
= ( ^ [F7: set @ ( product_prod @ A @ B ),F: A > B] :
( ! [A8: A,B7: B,B18: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ F7 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B18 ) @ F7 )
=> ( B7 = B18 ) ) )
& ! [A8: A,P3: $o] :
( ! [B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ F7 )
=> P3 )
=> P3 )
& ! [A8: A] :
( ( F @ A8 )
= ( the @ B
@ ^ [B7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B7 ) @ F7 ) ) ) ) ) ) ).
% su_rel_fun_def
thf(fact_3542_mult__cancel,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),X6: multiset @ A,Z7: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X6 @ Z7 ) @ ( plus_plus @ ( multiset @ A ) @ Y6 @ Z7 ) ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y6 ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel
thf(fact_3543_mult__cancel__add__mset,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),Uu2: A,X6: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu2 @ X6 ) @ ( add_mset @ A @ Uu2 @ Y6 ) ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y6 ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_3544_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A4: A,M4: multiset @ A] :
( ~ ( member @ A @ A4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M4 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_3545_Func__empty,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( insert2 @ ( A > B )
@ ^ [X2: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_3546_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A3: filter @ B,B2: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B2 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ A3 @ ( B2 @ I2 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_3547_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_3548_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_3549_mod__h__bot__normalize,axiom,
! [A: $tType,H2: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H2 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( 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_3550_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B2: filter @ B] :
( ( ( prod_filter @ A @ B @ A3 @ B2 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( filter @ A ) ) )
| ( B2
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_3551_one__step__implies__mult,axiom,
! [A: $tType,J4: multiset @ A,K5: multiset @ A,R3: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
( ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K5 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ 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 @ J4 ) ) @ ( mult @ A @ R3 ) ) ) ) ).
% one_step_implies_mult
thf(fact_3552_in__Inf__multiset__iff,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: A] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( member @ A @ X @ ( set_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) ) )
= ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ A3 )
=> ( member @ A @ X @ ( set_mset @ A @ X2 ) ) ) ) ) ) ).
% in_Inf_multiset_iff
thf(fact_3553_mult__implies__one__step,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),M4: multiset @ A,N3: multiset @ A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M4 @ N3 ) @ ( mult @ A @ R3 ) )
=> ? [I7: multiset @ A,J5: multiset @ A] :
( ( N3
= ( plus_plus @ ( multiset @ A ) @ I7 @ J5 ) )
& ? [K6: multiset @ A] :
( ( M4
= ( plus_plus @ ( multiset @ A ) @ I7 @ K6 ) )
& ( J5
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_mset @ A @ K6 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_3554_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 ) )
= ( ? [Q3: A > $o] :
( ( eventually @ A @ Q3 @ F5 )
& ! [X2: A,Y2: A] :
( ( Q3 @ X2 )
=> ( ( Q3 @ Y2 )
=> ( P @ ( product_Pair @ A @ A @ X2 @ Y2 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_3555_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F5: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F5 @ G6 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F5 )
& ( eventually @ B @ Pg @ G6 )
& ! [X2: A,Y2: B] :
( ( Pf @ X2 )
=> ( ( Pg @ Y2 )
=> ( P @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_3556_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B2: filter @ B,C2: filter @ A,D4: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B2
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A3 @ B2 ) @ ( prod_filter @ A @ B @ C2 @ D4 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A3 @ C2 )
& ( ord_less_eq @ ( filter @ B ) @ B2 @ D4 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_3557_infinite__set__mset__mset__set,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( set_mset @ A @ ( mset_set @ A @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_set_mset_mset_set
thf(fact_3558_mult__cancel__max,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),X6: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y6 ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X6 @ Y6 ) @ ( minus_minus @ ( multiset @ A ) @ Y6 @ X6 ) ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel_max
thf(fact_3559_set__mset__Inf,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( set_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ A3 ) ) ) ) ).
% set_mset_Inf
thf(fact_3560_set__mset__single,axiom,
! [A: $tType,B3: A] :
( ( set_mset @ A @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_mset_single
thf(fact_3561_eventually__prod2,axiom,
! [A: $tType,B: $tType,A3: filter @ A,P: B > $o,B2: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A] : P )
@ ( prod_filter @ A @ B @ A3 @ B2 ) )
= ( eventually @ B @ P @ B2 ) ) ) ).
% eventually_prod2
thf(fact_3562_eventually__prod1,axiom,
! [A: $tType,B: $tType,B2: filter @ A,P: B > $o,A3: filter @ B] :
( ( B2
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X2: B,Y2: A] : ( P @ X2 ) )
@ ( prod_filter @ B @ A @ A3 @ B2 ) )
= ( eventually @ B @ P @ A3 ) ) ) ).
% eventually_prod1
thf(fact_3563_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I4: set @ A,J4: set @ B,A3: A > ( filter @ C ),B2: B > ( filter @ D )] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A3 @ I4 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B2 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I2: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A3 @ I2 ) @ ( B2 @ J3 ) )
@ J4 ) )
@ I4 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_3564_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I4: set @ A,A3: A > ( filter @ B ),B2: filter @ C] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A3 @ I4 ) ) @ B2 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ ( A3 @ I2 ) @ B2 )
@ I4 ) ) ) ) ).
% prod_filter_INF1
thf(fact_3565_multp__code__iff__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > A > $o,N3: multiset @ A,M4: multiset @ A] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X3: A,Y3: A] :
( ( P @ X3 @ Y3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
=> ( ( multp_code @ A @ P @ N3 @ M4 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N3 @ M4 ) @ ( mult @ A @ R ) ) ) ) ) ) ).
% multp_code_iff_mult
thf(fact_3566_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > A > $o,N3: multiset @ A,M4: multiset @ A] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X3: A,Y3: A] :
( ( P @ X3 @ Y3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
=> ( ( multeqp_code @ A @ P @ N3 @ M4 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N3 @ M4 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
thf(fact_3567_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N5: multiset @ A,M5: multiset @ A] :
? [A5: A,M0: multiset @ A,K7: multiset @ A] :
( ( M5
= ( add_mset @ A @ A5 @ M0 ) )
& ( N5
= ( 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 ) @ R2 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_3568_mult1E,axiom,
! [A: $tType,N3: multiset @ A,M4: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N3 @ M4 ) @ ( mult1 @ A @ R3 ) )
=> ~ ! [A6: A,M02: multiset @ A] :
( ( M4
= ( add_mset @ A @ A6 @ M02 ) )
=> ! [K6: multiset @ A] :
( ( N3
= ( plus_plus @ ( multiset @ A ) @ M02 @ K6 ) )
=> ~ ! [B10: A] :
( ( member @ A @ B10 @ ( set_mset @ A @ K6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B10 @ A6 ) @ R3 ) ) ) ) ) ).
% mult1E
thf(fact_3569_mult1I,axiom,
! [A: $tType,M4: multiset @ A,A4: A,M03: multiset @ A,N3: multiset @ A,K5: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( M4
= ( add_mset @ A @ A4 @ M03 ) )
=> ( ( N3
= ( plus_plus @ ( multiset @ A ) @ M03 @ K5 ) )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( set_mset @ A @ K5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A4 ) @ R3 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N3 @ M4 ) @ ( mult1 @ A @ R3 ) ) ) ) ) ).
% mult1I
thf(fact_3570_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_3571_mult1__union,axiom,
! [A: $tType,B2: multiset @ A,D4: multiset @ A,R3: set @ ( product_prod @ A @ A ),C2: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B2 @ D4 ) @ ( mult1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ D4 ) ) @ ( mult1 @ A @ R3 ) ) ) ).
% mult1_union
thf(fact_3572_less__add,axiom,
! [A: $tType,N3: multiset @ A,A4: A,M03: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N3 @ ( add_mset @ A @ A4 @ M03 ) ) @ ( mult1 @ A @ R3 ) )
=> ( ? [M6: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M6 @ M03 ) @ ( mult1 @ A @ R3 ) )
& ( N3
= ( add_mset @ A @ A4 @ M6 ) ) )
| ? [K6: multiset @ A] :
( ! [B10: A] :
( ( member @ A @ B10 @ ( set_mset @ A @ K6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B10 @ A4 ) @ R3 ) )
& ( N3
= ( plus_plus @ ( multiset @ A ) @ M03 @ K6 ) ) ) ) ) ).
% less_add
thf(fact_3573_mult1__lessE,axiom,
! [A: $tType,N3: multiset @ A,M4: multiset @ A,R3: A > A > $o] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N3 @ M4 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) )
=> ( ( asymp @ A @ R3 )
=> ~ ! [A6: A,M02: multiset @ A] :
( ( M4
= ( add_mset @ A @ A6 @ M02 ) )
=> ! [K6: multiset @ A] :
( ( N3
= ( plus_plus @ ( multiset @ A ) @ M02 @ K6 ) )
=> ( ~ ( member @ A @ A6 @ ( set_mset @ A @ K6 ) )
=> ~ ! [B10: A] :
( ( member @ A @ B10 @ ( set_mset @ A @ K6 ) )
=> ( R3 @ B10 @ A6 ) ) ) ) ) ) ) ).
% mult1_lessE
thf(fact_3574_smsI,axiom,
! [A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat ),Z7: 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 ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ 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 ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ B2 ) ) @ ms_strict ) ) ).
% smsI
thf(fact_3575_wmsI,axiom,
! [A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat ),Z7: 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 ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ fun_max_strict )
| ( ( A3
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( B2
= ( 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 ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ B2 ) ) @ ms_weak ) ) ).
% wmsI
thf(fact_3576_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_3577_mult__cancel__max0,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),X6: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y6 ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X6 @ ( inter_mset @ A @ X6 @ Y6 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y6 @ ( inter_mset @ A @ X6 @ Y6 ) ) ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel_max0
thf(fact_3578_subset__mset_Oinf_Osemigroup__axioms,axiom,
! [A: $tType] : ( semigroup @ ( multiset @ A ) @ ( inter_mset @ A ) ) ).
% subset_mset.inf.semigroup_axioms
thf(fact_3579_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_3580_ms__weakI1,axiom,
! [Z7: multiset @ ( product_prod @ nat @ nat ),Z8: multiset @ ( product_prod @ nat @ nat ),A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z7 @ Z8 )
=> ( ( 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 ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ 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 ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z8 @ B2 ) ) @ ms_weak ) ) ) ).
% ms_weakI1
thf(fact_3581_ms__strictI,axiom,
! [Z7: multiset @ ( product_prod @ nat @ nat ),Z8: multiset @ ( product_prod @ nat @ nat ),A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z7 @ Z8 )
=> ( ( 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 ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ 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 ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z8 @ B2 ) ) @ ms_strict ) ) ) ).
% ms_strictI
thf(fact_3582_pw__leq__split,axiom,
! [X6: multiset @ ( product_prod @ nat @ nat ),Y6: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ X6 @ Y6 )
=> ? [A9: multiset @ ( product_prod @ nat @ nat ),B9: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
( ( X6
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A9 @ Z9 ) )
& ( Y6
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B9 @ Z9 ) )
& ( ( 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 ) @ A9 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B9 ) ) @ fun_max_strict )
| ( ( B9
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A9
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% pw_leq_split
thf(fact_3583_ms__weakI2,axiom,
! [Z7: multiset @ ( product_prod @ nat @ nat ),Z8: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z7 @ Z8 )
=> ( 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 ) ) @ Z7 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z8 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% ms_weakI2
thf(fact_3584_pw__leq_Ocases,axiom,
! [A1: multiset @ ( product_prod @ nat @ nat ),A22: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ A1 @ A22 )
=> ( ( ( A1
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
=> ( A22
!= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
=> ~ ! [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat,X8: multiset @ ( product_prod @ nat @ nat )] :
( ( A1
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( 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 ) @ Y3 @ ( 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 ) @ X3 @ Y3 ) @ fun_pair_leq )
=> ~ ( pw_leq @ X8 @ Y11 ) ) ) ) ) ) ).
% pw_leq.cases
thf(fact_3585_pw__leq__lstep,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( 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_3586_pw__leq__step,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X6: multiset @ ( product_prod @ nat @ nat ),Y6: 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 @ X6 @ Y6 )
=> ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X6 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y6 ) ) ) ) ).
% pw_leq_step
thf(fact_3587_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 ) ) ) ) )
| ? [X2: product_prod @ nat @ nat,Y2: product_prod @ nat @ nat,X7: multiset @ ( product_prod @ nat @ nat ),Y10: multiset @ ( product_prod @ nat @ nat )] :
( ( A12
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X7 ) )
& ( A23
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y10 ) )
& ( 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 @ X7 @ Y10 ) ) ) ) ) ).
% pw_leq.simps
thf(fact_3588_multp__def,axiom,
! [A: $tType] :
( ( multp @ A )
= ( ^ [R2: A > A > $o,M5: multiset @ A,N5: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M5 @ N5 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% multp_def
thf(fact_3589_prod__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,A3: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ A3 ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A3 )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% prod_mset.remove
thf(fact_3590_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_3591_subset__mset_OInf__fin_Oinsert,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A3 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.insert
thf(fact_3592_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,A4: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N @ A4 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ A4 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_3593_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_3594_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,N3: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( add_mset @ A @ X @ N3 ) )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ N3 ) ) ) ) ).
% prod_mset.add_mset
thf(fact_3595_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M4: multiset @ A,N3: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M4 @ N3 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M4 ) @ ( comm_m9189036328036947845d_mset @ A @ N3 ) ) ) ) ).
% prod_mset.union
thf(fact_3596_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ A,B2: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A3 ) @ ( comm_m9189036328036947845d_mset @ A @ B2 ) ) ) ) ).
% prod_mset_Un
thf(fact_3597_prod__mset_Oneutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ A3 ) )
=> ( X3
= ( one_one @ A ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod_mset.neutral
thf(fact_3598_subset__mset_OInf__fin_Ohom__commute,axiom,
! [A: $tType,H2: ( multiset @ A ) > ( multiset @ A ),N3: set @ ( multiset @ A )] :
( ! [X3: multiset @ A,Y3: multiset @ A] :
( ( H2 @ ( inter_mset @ A @ X3 @ Y3 ) )
= ( inter_mset @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H2 @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ N3 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H2 @ N3 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
thf(fact_3599_subset__mset_OInf__fin_Osubset,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B2 @ A3 )
=> ( ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B2 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset
thf(fact_3600_subset__mset_OInf__fin_Ounion,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B2 ) )
= ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B2 ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
thf(fact_3601_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: multiset @ A] :
( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A3 ) @ ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ A3 ) )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% is_unit_prod_mset_iff
thf(fact_3602_subset__mset_OcInf__eq__Inf__fin,axiom,
! [A: $tType,X6: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X6 )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ X6 ) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
thf(fact_3603_subset__mset_OInf__fin_Oclosed,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A,Y3: multiset @ A] : ( member @ ( multiset @ A ) @ ( inter_mset @ A @ X3 @ Y3 ) @ ( insert2 @ ( multiset @ A ) @ X3 @ ( insert2 @ ( multiset @ A ) @ Y3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ A3 ) ) ) ) ).
% subset_mset.Inf_fin.closed
thf(fact_3604_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ~ ( member @ ( multiset @ A ) @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A3 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_not_elem
thf(fact_3605_subset__mset_OInf__fin_Oremove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( member @ ( multiset @ A ) @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.remove
thf(fact_3606_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( 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 @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( 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 @ A3 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_remove
thf(fact_3607_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B2 ) )
= ( 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 @ A3 )
& ( member @ ( multiset @ A ) @ B4 @ B2 ) ) ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
thf(fact_3608_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) )
= ( 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 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf1_distrib
thf(fact_3609_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_3610_subset__mset_OInf__fin_OboundedE,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) )
=> ! [A10: multiset @ A] :
( ( member @ ( multiset @ A ) @ A10 @ A3 )
=> ( subseteq_mset @ A @ X @ A10 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
thf(fact_3611_subset__mset_OInf__fin_OboundedI,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [A6: multiset @ A] :
( ( member @ ( multiset @ A ) @ A6 @ A3 )
=> ( subseteq_mset @ A @ X @ A6 ) )
=> ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedI
thf(fact_3612_subset__mset_Osemilattice__neutr__axioms,axiom,
! [A: $tType] : ( semilattice_neutr @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.semilattice_neutr_axioms
thf(fact_3613_subset__mset_Osemigroup__axioms,axiom,
! [A: $tType] : ( semigroup @ ( multiset @ A ) @ ( union_mset @ A ) ) ).
% subset_mset.semigroup_axioms
thf(fact_3614_subset__mset_OcInf__eq__non__empty,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ A4 @ X3 ) )
=> ( ! [Y3: multiset @ A] :
( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ X6 )
=> ( subseteq_mset @ A @ Y3 @ X4 ) )
=> ( subseteq_mset @ A @ Y3 @ A4 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X6 )
= A4 ) ) ) ) ).
% subset_mset.cInf_eq_non_empty
thf(fact_3615_subset__mset_OcInf__greatest,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ Z2 @ X3 ) )
=> ( subseteq_mset @ A @ Z2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X6 ) ) ) ) ).
% subset_mset.cInf_greatest
thf(fact_3616_subset__mset_OcSup__eq__non__empty,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ X3 @ A4 ) )
=> ( ! [Y3: multiset @ A] :
( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ X6 )
=> ( subseteq_mset @ A @ X4 @ Y3 ) )
=> ( subseteq_mset @ A @ A4 @ Y3 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X6 )
= A4 ) ) ) ) ).
% subset_mset.cSup_eq_non_empty
thf(fact_3617_subset__mset_OcSup__least,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ X3 @ Z2 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ X6 ) @ Z2 ) ) ) ).
% subset_mset.cSup_least
thf(fact_3618_subset__mset_Ofinite__has__maximal,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ? [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A3 )
& ! [Xa2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Xa2 @ A3 )
=> ( ( subseteq_mset @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_3619_subset__mset_Ofinite__has__minimal,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ? [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A3 )
& ! [Xa2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Xa2 @ A3 )
=> ( ( subseteq_mset @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_3620_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A3: set @ B,M2: multiset @ A,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subseteq_mset @ A @ M2 @ ( F2 @ X3 ) ) )
=> ( subseteq_mset @ A @ M2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_3621_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A3: set @ B,F2: B > ( multiset @ A ),M4: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subseteq_mset @ A @ ( F2 @ X3 ) @ M4 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ M4 ) ) ) ).
% subset_mset.cSUP_least
thf(fact_3622_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ B2 )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B2 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
thf(fact_3623_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) )
= ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ A3 )
=> ( subseteq_mset @ A @ X @ X2 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.bounded_iff
thf(fact_3624_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) ) ) ) ).
% subset_mset.Inf_fin_le_Sup_fin
thf(fact_3625_subset__mset_OSup__fin_Oinsert__remove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( 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 @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( 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 @ A3 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.insert_remove
thf(fact_3626_subset__mset_OSup__fin_Oremove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( member @ ( multiset @ A ) @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A3 @ ( insert2 @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.remove
thf(fact_3627_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) )
= ( 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 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup1_distrib
thf(fact_3628_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_3629_subset__mset_OSup__fin_Oinsert,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A3 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Sup_fin.insert
thf(fact_3630_subset__mset_OSup__fin_OboundedE,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ X )
=> ! [A10: multiset @ A] :
( ( member @ ( multiset @ A ) @ A10 @ A3 )
=> ( subseteq_mset @ A @ A10 @ X ) ) ) ) ) ).
% subset_mset.Sup_fin.boundedE
thf(fact_3631_subset__mset_OSup__fin_OboundedI,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [A6: multiset @ A] :
( ( member @ ( multiset @ A ) @ A6 @ A3 )
=> ( subseteq_mset @ A @ A6 @ X ) )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ X ) ) ) ) ).
% subset_mset.Sup_fin.boundedI
thf(fact_3632_subset__mset_OSup__fin_Obounded__iff,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ X )
= ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ A3 )
=> ( subseteq_mset @ A @ X2 @ X ) ) ) ) ) ) ).
% subset_mset.Sup_fin.bounded_iff
thf(fact_3633_subset__mset_OcSup__eq__Sup__fin,axiom,
! [A: $tType,X6: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X6 )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ X6 ) ) ) ) ).
% subset_mset.cSup_eq_Sup_fin
thf(fact_3634_subset__mset_OSup__fin_Oclosed,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A,Y3: multiset @ A] : ( member @ ( multiset @ A ) @ ( union_mset @ A @ X3 @ Y3 ) @ ( insert2 @ ( multiset @ A ) @ X3 @ ( insert2 @ ( multiset @ A ) @ Y3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ A3 ) ) ) ) ).
% subset_mset.Sup_fin.closed
thf(fact_3635_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ~ ( member @ ( multiset @ A ) @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert2 @ ( multiset @ A ) @ X @ A3 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) ) ) ) ) ) ).
% subset_mset.Sup_fin.insert_not_elem
thf(fact_3636_subset__mset_OSup__fin_Osubset,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B2 @ A3 )
=> ( ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B2 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset
thf(fact_3637_subset__mset_OSup__fin_Ounion,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B2 ) )
= ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B2 ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.union
thf(fact_3638_subset__mset_OSup__fin_Ohom__commute,axiom,
! [A: $tType,H2: ( multiset @ A ) > ( multiset @ A ),N3: set @ ( multiset @ A )] :
( ! [X3: multiset @ A,Y3: multiset @ A] :
( ( H2 @ ( union_mset @ A @ X3 @ Y3 ) )
= ( union_mset @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ N3 )
=> ( ( N3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H2 @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ N3 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H2 @ N3 ) ) ) ) ) ) ).
% subset_mset.Sup_fin.hom_commute
thf(fact_3639_subset__mset_OSup__fin_Osubset__imp,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ B2 )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B2 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset_imp
thf(fact_3640_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( finite_finite @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite @ ( multiset @ A ) @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B2 ) )
= ( 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 @ A3 )
& ( member @ ( multiset @ A ) @ B4 @ B2 ) ) ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
thf(fact_3641_subset__mset_Omono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ! [F2: ( multiset @ A ) > B,A3: multiset @ A,B2: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A3 ) @ ( F2 @ B2 ) ) @ ( F2 @ ( union_mset @ A @ A3 @ B2 ) ) ) ) ) ).
% subset_mset.mono_sup
thf(fact_3642_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B2 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B2 ) ) ) ) ) ) ) ) ).
% subset_mset.cINF_union
thf(fact_3643_subset__mset_OcINF__insert,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( inter_mset @ A @ ( F2 @ A4 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) ) ) ) ) ).
% subset_mset.cINF_insert
thf(fact_3644_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B2 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B2 ) ) ) ) ) ) ) ) ).
% subset_mset.cSUP_union
thf(fact_3645_subset__mset_Obdd__above__empty,axiom,
! [A: $tType] : ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ).
% subset_mset.bdd_above_empty
thf(fact_3646_subset__mset_Obdd__below__empty,axiom,
! [A: $tType] : ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ).
% subset_mset.bdd_below_empty
thf(fact_3647_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_3648_subset__mset_OcInf__le__cSup,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.cInf_le_cSup
thf(fact_3649_subset__mset_OcSup__mono,axiom,
! [A: $tType,B2: set @ ( multiset @ A ),A3: set @ ( multiset @ A )] :
( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ! [B5: multiset @ A] :
( ( member @ ( multiset @ A ) @ B5 @ B2 )
=> ? [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ A3 )
& ( subseteq_mset @ A @ B5 @ X4 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ B2 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.cSup_mono
thf(fact_3650_subset__mset_OcSup__le__iff,axiom,
! [A: $tType,S: set @ ( multiset @ A ),A4: multiset @ A] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ S ) @ A4 )
= ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ S )
=> ( subseteq_mset @ A @ X2 @ A4 ) ) ) ) ) ) ).
% subset_mset.cSup_le_iff
thf(fact_3651_subset__mset_Ole__cInf__iff,axiom,
! [A: $tType,S: set @ ( multiset @ A ),A4: multiset @ A] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( subseteq_mset @ A @ A4 @ ( complete_Inf_Inf @ ( multiset @ A ) @ S ) )
= ( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ S )
=> ( subseteq_mset @ A @ A4 @ X2 ) ) ) ) ) ) ).
% subset_mset.le_cInf_iff
thf(fact_3652_subset__mset_OcInf__mono,axiom,
! [A: $tType,B2: set @ ( multiset @ A ),A3: set @ ( multiset @ A )] :
( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ! [B5: multiset @ A] :
( ( member @ ( multiset @ A ) @ B5 @ B2 )
=> ? [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ A3 )
& ( subseteq_mset @ A @ X4 @ B5 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B2 ) ) ) ) ) ).
% subset_mset.cInf_mono
thf(fact_3653_subset__mset_Omono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A3: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ ( multiset @ A ) @ B @ F2 @ A3 ) ) @ ( F2 @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) ) ) ) ) ) ) ).
% subset_mset.mono_cSup
thf(fact_3654_subset__mset_Omono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A3: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ ( multiset @ A ) @ B @ F2 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.mono_cInf
thf(fact_3655_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),U: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( subseteq_mset @ A @ ( F2 @ X2 ) @ U ) ) ) ) ) ) ).
% subset_mset.cSUP_le_iff
thf(fact_3656_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ B,G2: C > ( multiset @ A ),B2: set @ C,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G2 @ B2 ) )
=> ( ! [N4: B] :
( ( member @ B @ N4 @ A3 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B2 )
& ( subseteq_mset @ A @ ( F2 @ N4 ) @ ( G2 @ X4 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G2 @ B2 ) ) ) ) ) ) ).
% subset_mset.cSUP_mono
thf(fact_3657_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B2 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B2 ) ) @ ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B2 ) ) ) ) ) ) ).
% subset_mset.cSup_inter_less_eq
thf(fact_3658_subset__mset_Ole__cINF__iff,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),U: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( subseteq_mset @ A @ U @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( subseteq_mset @ A @ U @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% subset_mset.le_cINF_iff
thf(fact_3659_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B2: set @ B,F2: C > ( multiset @ A ),A3: set @ C,G2: B > ( multiset @ A )] :
( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B2 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A3 )
& ( subseteq_mset @ A @ ( F2 @ X4 ) @ ( G2 @ M3 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B2 ) ) ) ) ) ) ).
% subset_mset.cINF_mono
thf(fact_3660_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B2 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B2 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B2 ) ) ) ) ) ) ).
% subset_mset.less_eq_cInf_inter
thf(fact_3661_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A3: 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 ) @ A3 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I4 ) )
@ ( F2 @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I4 ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_3662_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A3: 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 ) @ A3 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I4 ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_3663_subset__mset_OcSup__subset__mono,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B2 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B2 )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B2 ) ) ) ) ) ).
% subset_mset.cSup_subset_mono
thf(fact_3664_subset__mset_OcInf__superset__mono,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B2 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B2 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ B2 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.cInf_superset_mono
thf(fact_3665_subset__mset_OcSup__cInf,axiom,
! [A: $tType,S: set @ ( multiset @ A )] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ S )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X2: multiset @ A] :
! [Y2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y2 @ S )
=> ( subseteq_mset @ A @ Y2 @ X2 ) ) ) ) ) ) ) ).
% subset_mset.cSup_cInf
thf(fact_3666_subset__mset_OcInf__cSup,axiom,
! [A: $tType,S: set @ ( multiset @ A )] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ S )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X2: multiset @ A] :
! [Y2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y2 @ S )
=> ( subseteq_mset @ A @ X2 @ Y2 ) ) ) ) ) ) ) ).
% subset_mset.cInf_cSup
thf(fact_3667_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,G2: B > ( multiset @ A ),B2: set @ B,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subseteq_mset @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B2 ) ) ) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
thf(fact_3668_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,G2: B > ( multiset @ A ),B2: set @ B,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( subseteq_mset @ A @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ B2 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.cINF_superset_mono
thf(fact_3669_subset__mset_OcSup__insert,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X6 ) )
= ( union_mset @ A @ A4 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X6 ) ) ) ) ) ).
% subset_mset.cSup_insert
thf(fact_3670_subset__mset_OcSup__insert__If,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A4: multiset @ A] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X6 ) )
= A4 ) )
& ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X6 ) )
= ( union_mset @ A @ A4 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X6 ) ) ) ) ) ) ).
% subset_mset.cSup_insert_If
thf(fact_3671_subset__mset_OcInf__insert__If,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A4: multiset @ A] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X6 ) )
= A4 ) )
& ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X6 ) )
= ( inter_mset @ A @ A4 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X6 ) ) ) ) ) ) ).
% subset_mset.cInf_insert_If
thf(fact_3672_subset__mset_OcInf__insert,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X6 ) )
= ( inter_mset @ A @ A4 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X6 ) ) ) ) ) ).
% subset_mset.cInf_insert
thf(fact_3673_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A3 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A3 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A5: B] : ( union_mset @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A3 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_3674_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A3 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G2 @ A3 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A5: B] : ( inter_mset @ A @ ( F2 @ A5 ) @ ( G2 @ A5 ) )
@ A3 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_3675_subset__mset_OcSup__union__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B2 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B2 ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B2 ) ) ) ) ) ) ) ).
% subset_mset.cSup_union_distrib
thf(fact_3676_subset__mset_OcInf__union__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B2: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B2 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B2 ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B2 ) ) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
thf(fact_3677_subset__mset_OcSUP__insert,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( union_mset @ A @ ( F2 @ A4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) ) ) ) ) ).
% subset_mset.cSUP_insert
thf(fact_3678_bdd__above__multiset__imp__finite__support,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( finite_finite @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( multiset @ A ) @ ( set @ A )
@ ^ [X7: multiset @ A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ X7 @ X2 ) ) )
@ A3 ) ) ) ) ) ).
% bdd_above_multiset_imp_finite_support
thf(fact_3679_Sup__multiset__in__multiset,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( finite_finite @ A
@ ( collect @ A
@ ^ [I2: A] :
( ord_less @ nat @ ( zero_zero @ nat )
@ ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [M5: multiset @ A] : ( count @ A @ M5 @ I2 )
@ A3 ) ) ) ) ) ) ) ).
% Sup_multiset_in_multiset
thf(fact_3680_count__Sup__multiset__nonempty,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: A] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( count @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ X )
= ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X7: multiset @ A] : ( count @ A @ X7 @ X )
@ A3 ) ) ) ) ) ).
% count_Sup_multiset_nonempty
thf(fact_3681_subset__mset_OatLeastAtMost__singleton__iff,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,C3: multiset @ A] :
( ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 )
= ( insert2 @ ( multiset @ A ) @ C3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( ( A4 = B3 )
& ( B3 = C3 ) ) ) ).
% subset_mset.atLeastAtMost_singleton_iff
thf(fact_3682_subset__mset_OatLeastatMost__empty__iff2,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( ( bot_bot @ ( set @ ( multiset @ A ) ) )
= ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 ) )
= ( ~ ( subseteq_mset @ A @ A4 @ B3 ) ) ) ).
% subset_mset.atLeastatMost_empty_iff2
thf(fact_3683_subset__mset_OatLeastatMost__empty__iff,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( ~ ( subseteq_mset @ A @ A4 @ B3 ) ) ) ).
% subset_mset.atLeastatMost_empty_iff
thf(fact_3684_subset__mset_OatLeastAtMost__singleton,axiom,
! [A: $tType,A4: multiset @ A] :
( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ A4 )
= ( insert2 @ ( multiset @ A ) @ A4 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastAtMost_singleton
thf(fact_3685_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X: set @ ( multiset @ A )] :
( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ X ) )
= ( ^ [I2: 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 @ I2 )
@ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X ) ) ) ) ) ) ).
% Inf_multiset.rep_eq
thf(fact_3686_count__Inf__multiset__nonempty,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: A] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ X )
= ( complete_Inf_Inf @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X7: multiset @ A] : ( count @ A @ X7 @ X )
@ A3 ) ) ) ) ).
% count_Inf_multiset_nonempty
thf(fact_3687_subset__mset_OatLeastAtMost__singleton_H,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( A4 = B3 )
=> ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 )
= ( insert2 @ ( multiset @ A ) @ A4 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.atLeastAtMost_singleton'
thf(fact_3688_Sup__multiset__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) )
= ( ^ [A8: set @ ( multiset @ A )] :
( if @ ( multiset @ A )
@ ( ( A8
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A8 ) )
@ ( abs_multiset @ A
@ ^ [I2: A] :
( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X7: multiset @ A] : ( count @ A @ X7 @ I2 )
@ A8 ) ) )
@ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Sup_multiset_def
thf(fact_3689_count__image__mset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: multiset @ B,X: A] :
( ( count @ A @ ( image_mset @ B @ A @ F2 @ A3 ) @ X )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ A3 ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( set_mset @ B @ A3 ) ) ) ) ).
% count_image_mset
thf(fact_3690_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C3: A,A3: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( Y = X2 ) @ C3 @ ( zero_zero @ A ) )
@ A3 ) )
= ( times_times @ A @ C3 @ ( semiring_1_of_nat @ A @ ( count @ B @ A3 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_3691_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C3: A,A3: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( X2 = Y ) @ C3 @ ( zero_zero @ A ) )
@ A3 ) )
= ( times_times @ A @ C3 @ ( semiring_1_of_nat @ A @ ( count @ B @ A3 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_3692_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A3 ) )
= ( one_one @ A ) ) ) ).
% prod_mset.neutral_const
thf(fact_3693_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,X: B,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ ( add_mset @ B @ X @ A3 ) ) )
= ( times_times @ A @ ( G2 @ X ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ A3 ) ) ) ) ) ).
% prod_mset.insert
thf(fact_3694_sum__mset__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( times @ A )
& ( semiring_0 @ B ) )
=> ! [F2: A > B,A3: multiset @ A,G2: C > B,B2: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F2 @ A3 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G2 @ B2 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I2: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I2 ) @ ( G2 @ J3 ) )
@ B2 ) )
@ A3 ) ) ) ) ).
% sum_mset_product
thf(fact_3695_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,M4: multiset @ B,C3: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F2 @ M4 ) ) @ C3 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C3 )
@ M4 ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_3696_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C3: A,F2: B > A,M4: multiset @ B] :
( ( times_times @ A @ C3 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F2 @ M4 ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C3 @ ( F2 @ X2 ) )
@ M4 ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_3697_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,H2: B > A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G2 @ X2 ) @ ( H2 @ X2 ) )
@ A3 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ A3 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H2 @ A3 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_3698_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] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M4 ) )
=> ( ( F2 @ X3 @ Y3 )
= ( G2 @ X3 @ Y3 ) ) )
=> ( ( 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_3699_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C3: A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( X2 = Y ) @ C3 @ ( one_one @ A ) )
@ A3 ) )
= ( power_power @ A @ C3 @ ( count @ B @ A3 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_3700_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C3: A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( Y = X2 ) @ C3 @ ( one_one @ A ) )
@ A3 ) )
= ( power_power @ A @ C3 @ ( count @ B @ A3 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_3701_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ B,B2: multiset @ B,G2: B > A] :
( ( ( inter_mset @ B @ A3 @ B2 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ ( union_mset @ B @ A3 @ B2 ) ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ A3 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G2 @ B2 ) ) ) ) ) ) ).
% prod_mset.union_disjoint
thf(fact_3702_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 )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I2 )
@ A8 ) ) ) ) ) ).
% Inf_multiset_def
thf(fact_3703_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [Y: B,A3: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [X2: A] : Y
@ A3 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( multiset @ A ) @ A3 ) ) @ Y ) ) ) ).
% sum_mset_constant
thf(fact_3704_relpow__fun__conv,axiom,
! [A: $tType,A4: A,B3: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
= ( ? [F: nat > A] :
( ( ( F @ ( zero_zero @ nat ) )
= A4 )
& ( ( F @ N )
= B3 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_3705_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I3: code_integer] :
( X
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I3 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_3706_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A ),X4: A,Y5: A,Z6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z6 ) @ R ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ W ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_3707_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_3708_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_3709_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 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_3710_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N: nat,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( 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_3711_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 ) )
=> ? [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_3712_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_3713_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( 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_3714_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_3715_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 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_3716_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 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_3717_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_3718_exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I3: int] :
( X
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I3 ) ) ) ).
% exhaustive_int'.cases
thf(fact_3719_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 )] :
~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I3: int] :
( X
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I3 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_3720_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 )] :
~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I3: 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 ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I3 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_3721_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_finite @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) )
@ X
@ X )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ ( A > nat ) @ ( multiset @ A ) @ ( abs_multiset @ A ) @ X ) )
= ( abs_multiset @ A
@ ^ [I2: A] :
( if @ nat
@ ( X
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I2 )
@ X ) ) ) ) ) ) ).
% Inf_multiset.abs_eq
thf(fact_3722_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_3723_equivp__equiv,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A3 )
= ( equiv_equivp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ A3 ) ) ) ).
% equivp_equiv
thf(fact_3724_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_3725_asym_Ocases,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A4 )
=> ! [A10: A,B10: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A10 @ B10 ) @ A4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B10 @ A10 ) @ A4 ) ) ) ).
% asym.cases
thf(fact_3726_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A5: set @ ( product_prod @ A @ A )] :
? [R6: set @ ( product_prod @ A @ A )] :
( ( A5 = R6 )
& ! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R6 ) ) ) ) ) ).
% asym.simps
thf(fact_3727_asym_Ointros,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R ) )
=> ( asym @ A @ R ) ) ).
% asym.intros
thf(fact_3728_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_3729_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
! [X2: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R6 ) ) ) ) ).
% asym_iff
thf(fact_3730_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_finite @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F: A > nat] :
( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I2 )
@ A8 ) ) )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I2 )
@ A8 ) ) ) ) ).
% Inf_multiset.rsp
thf(fact_3731_empty__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_set @ A @ B @ A3 @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ B ) ) ) ).
% empty_transfer
thf(fact_3732_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 )
& ~ ? [X7: B > A] : ( bNF_Wellorder_embed @ B @ A @ R5 @ R3 @ X7 ) ) ) ).
% ordLess_iff
thf(fact_3733_Range__insert,axiom,
! [A: $tType,B: $tType,A4: B,B3: A,R3: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B3 ) @ R3 ) )
= ( insert2 @ A @ B3 @ ( range2 @ B @ A @ R3 ) ) ) ).
% Range_insert
thf(fact_3734_Range__empty,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Range_empty
thf(fact_3735_Range__iff,axiom,
! [A: $tType,B: $tType,A4: A,R3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A4 @ ( range2 @ B @ A @ R3 ) )
= ( ? [Y2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y2 @ A4 ) @ R3 ) ) ) ).
% Range_iff
thf(fact_3736_RangeE,axiom,
! [A: $tType,B: $tType,B3: A,R3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B3 @ ( range2 @ B @ A @ R3 ) )
=> ~ ! [A6: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B3 ) @ R3 ) ) ).
% RangeE
thf(fact_3737_Range_Ointros,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 )
=> ( member @ B @ B3 @ ( range2 @ A @ B @ R3 ) ) ) ).
% Range.intros
thf(fact_3738_Range_Osimps,axiom,
! [B: $tType,A: $tType,A4: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A4 @ ( range2 @ A @ B @ R3 ) )
= ( ? [A5: A,B4: B] :
( ( A4 = B4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R3 ) ) ) ) ).
% Range.simps
thf(fact_3739_Range_Ocases,axiom,
! [B: $tType,A: $tType,A4: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A4 @ ( range2 @ A @ B @ R3 ) )
=> ~ ! [A6: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A4 ) @ R3 ) ) ).
% Range.cases
thf(fact_3740_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_3741_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_3742_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: 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 ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R1 @ R22 @ F2 )
=> ( ( image2 @ A @ B @ F2 @ ( field2 @ A @ R1 ) )
!= ( field2 @ B @ R22 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_3743_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),F13: A > C,F23: 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 ) ) @ R1 @ R22 ) @ ( 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 ) ) @ R22 @ R32 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R1 @ R32 @ F13 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R22 @ R32 @ F23 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F13 @ ( field2 @ A @ R1 ) ) @ ( image2 @ B @ C @ F23 @ ( field2 @ B @ R22 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R32 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_3744_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
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F: A > nat] : ( F @ I2 )
@ A8 ) ) )
@ ( complete_Inf_Inf @ ( multiset @ A ) ) ) ).
% Inf_multiset.transfer
thf(fact_3745_wf__UN,axiom,
! [B: $tType,A: $tType,I4: set @ A,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ( wf @ B @ ( R3 @ I3 ) ) )
=> ( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I4 )
=> ( ( member @ A @ J2 @ I4 )
=> ( ( ( R3 @ I3 )
!= ( R3 @ J2 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R3 @ I3 ) ) @ ( 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_3746_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_3747_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_3748_Domain__insert,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 ) )
= ( insert2 @ A @ A4 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain_insert
thf(fact_3749_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A4: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B5 ) @ R3 ) ) ).
% Domain.cases
thf(fact_3750_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A4: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R3 ) )
= ( ? [A5: A,B4: B] :
( ( A4 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_3751_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R3 )
=> ( member @ A @ A4 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_3752_DomainE,axiom,
! [B: $tType,A: $tType,A4: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B5 ) @ R3 ) ) ).
% DomainE
thf(fact_3753_Domain__iff,axiom,
! [A: $tType,B: $tType,A4: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R3 ) )
= ( ? [Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ Y2 ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_3754_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A ),Y: A] :
( ~ ( member @ A @ X @ ( domain @ A @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_3755_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_3756_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X2: A] :
? [Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 ) ) ) ) ).
% Domain_unfold
thf(fact_3757_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_3758_wf__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( R
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M3: A] :
~ ( member @ A @ M3 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range2 @ A @ A @ R ) ) ) ) ) ).
% wf_min
thf(fact_3759_wf__Un,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( wf @ A @ S2 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R3 ) @ ( range2 @ A @ A @ S2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_3760_wf__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
=> ( wf @ A @ X3 ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
=> ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ X3 ) @ ( range2 @ A @ A @ Xa3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( wf @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) ) ) ).
% wf_Union
thf(fact_3761_wf__max,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( converse @ A @ A @ R ) )
=> ( ( R
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M3: A] :
~ ( member @ A @ M3 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R ) @ ( domain @ A @ A @ R ) ) ) ) ) ).
% wf_max
thf(fact_3762_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( domain @ A @ B @ R ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R @ X ) ) @ R ) ) ).
% for_in_RI
thf(fact_3763_ord__to__filter__compat,axiom,
! [A: $tType,R0: set @ ( product_prod @ A @ A )] :
( bNF_Wellorder_compat @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A )
@ ( 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_3764_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 ) ) ) ) ) ).
% Range_def
thf(fact_3765_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: B] : ( member @ B @ X2 @ ( range2 @ A @ B @ R3 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_3766_compat__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_compat @ A @ A2 )
= ( ^ [R2: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ A2 @ A2 ),F: A > A2] :
! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R2 )
=> ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F @ A5 ) @ ( F @ B4 ) ) @ R8 ) ) ) ) ).
% compat_def
thf(fact_3767_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% in_chain_finite
thf(fact_3768_cofinite__bot,axiom,
! [A: $tType] :
( ( ( cofinite @ A )
= ( bot_bot @ ( filter @ A ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% cofinite_bot
thf(fact_3769_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_3770_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_3771_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_3772_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,N: nat] :
( ( divide_divide @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A4 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_3773_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_3774_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_3775_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_3776_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F: A > B,R6: set @ ( product_prod @ A @ A ),X2: A,Y2: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R6 ) @ ( F @ Y2 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_3777_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( minus_minus @ ( set @ A ) @ S7 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R6 @ X2 @ Y2 ) ) ) ) ) ).
% pairwise_alt
thf(fact_3778_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_3779_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_3780_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A4 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_3781_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A4: A,R: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A4 ) @ R )
=> ( ( cut @ A @ B @ F2 @ R @ A4 @ X )
= ( F2 @ X ) ) ) ).
% cut_apply
thf(fact_3782_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 ) )
= ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R )
=> ( ( F2 @ Y2 )
= ( G2 @ Y2 ) ) ) ) ) ).
% cuts_eq
thf(fact_3783_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A3: A] : ( pairwise @ A @ P @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_3784_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_3785_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R2 ) ) ) ) ).
% above_def
thf(fact_3786_ID_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( bNF_id_bnf @ ( A > B > $o ) )
= ( ^ [R6: A > B > $o,A5: A,B4: B] :
? [Z3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ Z3
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( insert2 @ ( product_prod @ A @ B ) @ X2 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > A ) @ ( product_fst @ A @ B ) @ Z3 )
= A5 )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > B ) @ ( product_snd @ A @ B ) @ Z3 )
= B4 ) ) ) ) ).
% ID.in_rel
thf(fact_3787_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A5: A] :
( ( A5
= ( one_one @ A ) )
| ? [X2: A] :
( ( A5
= ( uminus_uminus @ A @ X2 ) )
& ( neg_numeral_is_num @ A @ X2 ) )
| ? [X2: A,Y2: A] :
( ( A5
= ( plus_plus @ A @ X2 @ Y2 ) )
& ( neg_numeral_is_num @ A @ X2 )
& ( neg_numeral_is_num @ A @ Y2 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_3788_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_3789_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra2: A > A > $o] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( Ra2 @ Z4 @ Z4 ) )
=> ( bNF_id_bnf @ ( A > A > $o ) @ Ra2 @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_3790_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: A,Y: B,Ra2: A > B > $o] :
( ( bNF_id_bnf @ ( A > B > $o ) @ R @ X @ Y )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member @ B @ Yb @ ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_id_bnf @ ( A > B > $o ) @ Ra2 @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_3791_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya2: A,Y: B,Xa: B,R: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya2 )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( insert2 @ A @ Ya2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member @ B @ Yb @ ( insert2 @ B @ Xa @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_id_bnf @ ( A > B > $o ) @ R @ X @ Y )
= ( bNF_id_bnf @ ( A > B > $o ) @ Ra2 @ Ya2 @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_3792_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A4: A] :
( ( neg_numeral_is_num @ A @ A4 )
=> ( ( A4
!= ( one_one @ A ) )
=> ( ! [X3: A] :
( ( A4
= ( uminus_uminus @ A @ X3 ) )
=> ~ ( neg_numeral_is_num @ A @ X3 ) )
=> ~ ! [X3: A,Y3: A] :
( ( A4
= ( plus_plus @ A @ X3 @ Y3 ) )
=> ( ( neg_numeral_is_num @ A @ X3 )
=> ~ ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_3793_subset__mset_OatLeastatMost__empty,axiom,
! [A: $tType,B3: multiset @ A,A4: multiset @ A] :
( ( subset_mset @ A @ B3 @ A4 )
=> ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastatMost_empty
thf(fact_3794_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ A4 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B3 ) ) )
= ( divide_divide @ A @ A4 @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_3795_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ A4 @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_3796_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B3 @ A4 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B3 ) @ A4 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_3797_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_3798_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_3799_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A4: A,B3: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ).
% unit_factor_mult
thf(fact_3800_subset__mset_OacyclicI__order,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ B ),F2: B > ( multiset @ A )] :
( ! [A6: B,B5: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B5 ) @ R3 )
=> ( subset_mset @ A @ ( F2 @ B5 ) @ ( F2 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R3 ) ) ).
% subset_mset.acyclicI_order
thf(fact_3801_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A4 )
= A4 ) ) ) ).
% is_unit_unit_factor
thf(fact_3802_gcd__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A4: A,B3: A] :
( ( times_times @ A @ K @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K @ A4 ) @ ( times_times @ A @ K @ B3 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% gcd_mult_distrib
thf(fact_3803_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ C3 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) @ ( unit_f5069060285200089521factor @ A @ C3 ) ) ) ) ).
% mult_gcd_right
thf(fact_3804_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( times_times @ A @ C3 @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C3 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_3805_subset__implies__mult,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A3 @ B2 ) @ ( mult @ A @ R3 ) ) ) ).
% subset_implies_mult
thf(fact_3806_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_3807_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( ( A4
= ( zero_zero @ A ) )
& ( B3
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A4
= ( zero_zero @ A ) )
& ( B3
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_3808_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B3: A,D3: A,A4: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B3 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( ( ( times_times @ A @ A4 @ D3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( ( A4 = C3 )
& ( B3 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_3809_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( ( gcd_Gcd @ A @ A3 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_3810_subset__mset_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A4 @ B3 )
= ( minus_minus @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 ) @ ( insert2 @ ( multiset @ A ) @ A4 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_3811_subset__mset_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A4 @ B3 )
= ( minus_minus @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 ) @ ( insert2 @ ( multiset @ A ) @ B3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3812_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_3813_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_3814_subset__mset_OatLeastLessThan__empty,axiom,
! [A: $tType,B3: multiset @ A,A4: multiset @ A] :
( ( subseteq_mset @ A @ B3 @ A4 )
=> ( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastLessThan_empty
thf(fact_3815_subset__mset_OatLeastLessThan__empty__iff,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( ~ ( subset_mset @ A @ A4 @ B3 ) ) ) ).
% subset_mset.atLeastLessThan_empty_iff
thf(fact_3816_subset__mset_OatLeastLessThan__empty__iff2,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A] :
( ( ( bot_bot @ ( set @ ( multiset @ A ) ) )
= ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A4 @ B3 ) )
= ( ~ ( subset_mset @ A @ A4 @ B3 ) ) ) ).
% subset_mset.atLeastLessThan_empty_iff2
thf(fact_3817_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_3818_subset__mset_OgreaterThanAtMost__empty__iff,axiom,
! [A: $tType,K: multiset @ A,L: multiset @ A] :
( ( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ K @ L )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( ~ ( subset_mset @ A @ K @ L ) ) ) ).
% subset_mset.greaterThanAtMost_empty_iff
thf(fact_3819_subset__mset_OgreaterThanAtMost__empty__iff2,axiom,
! [A: $tType,K: multiset @ A,L: multiset @ A] :
( ( ( bot_bot @ ( set @ ( multiset @ A ) ) )
= ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ K @ L ) )
= ( ~ ( subset_mset @ A @ K @ L ) ) ) ).
% subset_mset.greaterThanAtMost_empty_iff2
thf(fact_3820_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I4: set @ B,F2: B > ( multiset @ A )] :
( ( finite_finite @ B @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I4 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F2 @ I3 ) ) )
=> ( 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_3821_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),G2: B > ( multiset @ A )] :
( ( finite_finite @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subset_mset @ A @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F2 @ A3 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G2 @ A3 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_3822_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
thf(fact_3823_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A8: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A8
@ ( collect @ A
@ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_3824_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ ( normal6383669964737779283malize @ A @ B3 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_3825_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_3826_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_3827_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_3828_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_3829_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( gcd_Lcm @ A @ A3 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_3830_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= A4 ) ) ).
% normalize_mult_unit_factor
thf(fact_3831_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ A4 ) )
= A4 ) ) ).
% unit_factor_mult_normalize
thf(fact_3832_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_3833_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_3834_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_3835_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_3836_Lcm__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% Lcm_singleton
thf(fact_3837_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ A4 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A4 ) ) ) ) ).
% normalize_div
thf(fact_3838_Gcd__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% Gcd_singleton
thf(fact_3839_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A4: A,B3: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% normalize_mult
thf(fact_3840_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A,C3: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Lcm @ A @ A3 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_3841_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= A4 )
=> ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
= ( A4
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_3842_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_3843_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_3844_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
=> ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_3845_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C3 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_3846_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B3 @ A4 ) @ C3 ) ) ) ) ).
% gcd_mult_right
thf(fact_3847_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_3848_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,D3: A,B3: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B3 @ C3 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ C3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B3 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
& ( ( normal6383669964737779283malize @ A @ C3 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_3849_Lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,B5: A] :
( ( member @ A @ A6 @ A3 )
=> ( ( member @ A @ B5 @ A3 )
=> ( ( A6 != B5 )
=> ( algebr8660921524188924756oprime @ A @ A6 @ B5 ) ) ) )
=> ( ( gcd_Lcm @ A @ A3 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ A3 ) ) ) ) ) ) ) ).
% Lcm_coprime
thf(fact_3850_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_3851_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C3: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ) ).
% Gcd_mult
thf(fact_3852_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( ( gcd_Lcm @ A @ A3 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_3853_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A,B3: A] :
( ( finite_finite @ A @ A3 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B3 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B3 @ ( semiring_gcd_Gcd_fin @ A @ A3 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_3854_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% gcd.bounded_quasi_semilattice_axioms
thf(fact_3855_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A,B3: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B3 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B3 @ ( semiring_gcd_Lcm_fin @ A @ A3 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_3856_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A4 @ B3 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ ( gcd_lcm @ A @ A4 @ B3 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_3857_Lcm__eq__Max__nat,axiom,
! [M4: set @ nat] :
( ( finite_finite @ nat @ M4 )
=> ( ( M4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ! [M3: nat,N4: nat] :
( ( member @ nat @ M3 @ M4 )
=> ( ( member @ nat @ N4 @ M4 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N4 ) @ M4 ) ) )
=> ( ( gcd_Lcm @ nat @ M4 )
= ( lattic643756798349783984er_Max @ nat @ M4 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_3858_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( gcd_lcm @ A @ A4 @ B3 )
= ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% lcm_eq_1_iff
thf(fact_3859_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A] :
( ( gcd_lcm @ A @ A4 @ ( one_one @ A ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% lcm.top_right_normalize
thf(fact_3860_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A] :
( ( gcd_lcm @ A @ ( one_one @ A ) @ A4 )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% lcm.top_left_normalize
thf(fact_3861_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_3862_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A3 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_3863_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( ( A4
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A4
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_3864_lcm__mult__gcd,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% lcm_mult_gcd
thf(fact_3865_gcd__mult__lcm,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% gcd_mult_lcm
thf(fact_3866_Lcm__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,B3: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_lcm @ A @ A4 @ B3 ) ) ) ).
% Lcm_2
thf(fact_3867_lcm_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( semigroup @ A @ ( gcd_lcm @ A ) ) ) ).
% lcm.semigroup_axioms
thf(fact_3868_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( gcd_lcm @ A @ A4 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Lcm_fin.insert_remove
thf(fact_3869_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A3 )
= ( gcd_lcm @ A @ A4 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Lcm_fin.remove
thf(fact_3870_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% lcm.bounded_quasi_semilattice_axioms
thf(fact_3871_Lcm__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% Lcm_fin_def
thf(fact_3872_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C3 ) @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_3873_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C3: A,B3: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B3 @ A4 ) @ C3 ) ) ) ) ).
% lcm_mult_right
thf(fact_3874_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_lcm @ A @ A4 @ B3 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_3875_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A8: set @ A] : ( if @ A @ ( finite_finite @ A @ A8 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_3876_lcm__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A4: A,B3: A] :
( ( times_times @ A @ K @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K @ A4 ) @ ( times_times @ A @ K @ B3 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% lcm_mult_distrib
thf(fact_3877_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ C3 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A4 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) @ ( unit_f5069060285200089521factor @ A @ C3 ) ) ) ) ).
% mult_lcm_right
thf(fact_3878_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A4: A,B3: A] :
( ( times_times @ A @ C3 @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C3 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A4 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_3879_Lcm__in__lcm__closed__set__nat,axiom,
! [M4: set @ nat] :
( ( finite_finite @ nat @ M4 )
=> ( ( M4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ! [M3: nat,N4: nat] :
( ( member @ nat @ M3 @ M4 )
=> ( ( member @ nat @ N4 @ M4 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N4 ) @ M4 ) ) )
=> ( member @ nat @ ( gcd_Lcm @ nat @ M4 ) @ M4 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_3880_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B3 @ ( times_times @ A @ C3 @ A4 ) )
= ( gcd_lcm @ A @ B3 @ C3 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_3881_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B3 @ A4 ) @ C3 )
= ( gcd_lcm @ A @ B3 @ C3 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_3882_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B3 @ A4 ) @ C3 )
= ( gcd_lcm @ A @ B3 @ C3 ) ) ) ) ).
% lcm_div_unit1
thf(fact_3883_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B3 @ ( divide_divide @ A @ C3 @ A4 ) )
= ( gcd_lcm @ A @ B3 @ C3 ) ) ) ) ).
% lcm_div_unit2
thf(fact_3884_lcm__gcd__prod,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% lcm_gcd_prod
thf(fact_3885_lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( gcd_lcm @ A @ A4 @ B3 )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% lcm_coprime
thf(fact_3886_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A3 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) )
& ( finite_finite @ A @ A3 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_3887_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_3888_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
thf(fact_3889_in__range_Osimps,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ As )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% in_range.simps
thf(fact_3890_in__range_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( Y
= ( ~ ! [X2: nat] :
( ( member @ nat @ X2 @ As3 )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_3891_in__range_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ~ ! [X4: nat] :
( ( member @ nat @ X4 @ As3 )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_3892_in__range_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ! [X3: nat] :
( ( member @ nat @ X3 @ As3 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_3893_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,X: A,H2: heap_ext @ product_unit,As: set @ nat] :
( ( sngr_assn_raw @ A @ R3 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ( ( ref_get @ A @ H2 @ R3 )
= X )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ R3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ R3 ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% sngr_assn_raw.simps
thf(fact_3894_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( Y
= ( ~ ( ( ( ref_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
thf(fact_3895_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ~ ( ( ( ref_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(2)
thf(fact_3896_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( ( ref_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(3)
thf(fact_3897_relH__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit,R3: ref @ A] :
( ( relH @ As @ H2 @ H5 )
=> ( ( member @ nat @ ( addr_of_ref @ A @ R3 ) @ As )
=> ( ( ref_get @ A @ H2 @ R3 )
= ( ref_get @ A @ H5 @ R3 ) ) ) ) ) ).
% relH_ref
thf(fact_3898_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( Y
= ( ( ( ref_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
thf(fact_3899_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) )
=> ~ ( ( ( ref_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
thf(fact_3900_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) )
=> ( ( ( ref_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
thf(fact_3901_relH__set__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,As: set @ nat,H2: heap_ext @ product_unit,X: A] :
( ~ ( member @ nat @ ( addr_of_ref @ A @ R3 ) @ As )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( relH @ As @ H2 @ ( ref_set @ A @ R3 @ X @ H2 ) ) ) ) ) ).
% relH_set_ref
thf(fact_3902_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: array @ A,X: list @ A,H2: heap_ext @ product_unit,As: set @ nat] :
( ( snga_assn_raw @ A @ R3 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ( ( array_get @ A @ H2 @ R3 )
= X )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ R3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ R3 ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% snga_assn_raw.simps
thf(fact_3903_snga__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( Y
= ( ~ ( ( ( array_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
thf(fact_3904_snga__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ~ ( ( ( array_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ).
% snga_assn_raw.elims(2)
thf(fact_3905_snga__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( ( array_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ).
% snga_assn_raw.elims(3)
thf(fact_3906_relH__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As: set @ nat,H2: heap_ext @ product_unit,H5: heap_ext @ product_unit,R3: array @ A] :
( ( relH @ As @ H2 @ H5 )
=> ( ( member @ nat @ ( addr_of_array @ A @ R3 ) @ As )
=> ( ( array_get @ A @ H2 @ R3 )
= ( array_get @ A @ H5 @ R3 ) ) ) ) ) ).
% relH_array
thf(fact_3907_snga__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( Y
= ( ( ( array_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
thf(fact_3908_snga__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) )
=> ~ ( ( ( array_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
thf(fact_3909_snga__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) )
=> ( ( ( array_get @ A @ H3 @ X )
= Xa )
& ( As3
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
thf(fact_3910_relH__set__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: array @ A,As: set @ nat,H2: heap_ext @ product_unit,X: list @ A] :
( ~ ( member @ nat @ ( addr_of_array @ A @ R3 ) @ As )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( relH @ As @ H2 @ ( array_set @ A @ R3 @ X @ H2 ) ) ) ) ) ).
% relH_set_array
thf(fact_3911_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),As8: A > B] :
! [I2: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J3 ) @ R2 )
=> ( ord_less_eq @ B @ ( As8 @ I2 ) @ ( As8 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_3912_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_3913_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A8: 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 @ A8 ) ) ) ) ) ).
% Gr_def
thf(fact_3914_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T4: set @ C,H2: B > C,S: set @ B,T5: set @ C,G2: C > A] :
( ( finite_finite @ B @ S6 )
=> ( ( finite_finite @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S6 )
=> ( ( G2 @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T4 )
=> ( ( G2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G2 @ ( H2 @ X2 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G2 @ T5 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_3915_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) @ A3 )
=> ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_3916_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( bij_betw @ A @ B @ F2 @ A3 @ ( bot_bot @ ( set @ B ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_3917_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A3: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) )
=> ( member @ A @ X @ A3 ) ) ).
% GrD1
thf(fact_3918_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A3: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) )
=> ( ( F2 @ X )
= Fx ) ) ).
% GrD2
thf(fact_3919_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A3: set @ A,F2: A > B,A17: set @ B] :
( ~ ( member @ A @ B3 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B3 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A17 )
= ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert2 @ B @ ( F2 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_3920_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A3: set @ A,F2: A > B,A17: set @ B] :
( ~ ( member @ A @ B3 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B3 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A17 )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert2 @ B @ ( F2 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_3921_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B2: set @ B,C2: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ B2 )
=> ( ( bij_betw @ A @ B @ F2 @ C2 @ D4 )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D4 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_3922_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C2: set @ A,B2: set @ B,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D4 ) )
=> ( ( bij_betw @ A @ B @ F2 @ C2 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ C2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ A3 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_3923_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),F2: A > B,A17: set @ B] :
( ( order_well_order_on @ A @ A3 @ R3 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A17 )
=> ? [R9: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A17 @ R9 )
& ( 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 @ R9 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_3924_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C2: set @ B,G2: A > B,B2: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ C2 )
=> ( ( bij_betw @ A @ B @ G2 @ B2 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C2 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A3 ) @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B2 )
@ ( sup_sup @ ( set @ B ) @ C2 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_3925_infinite__imp__bij__betw2,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ~ ( finite_finite @ A @ A3 )
=> ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_3926_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ~ ( finite_finite @ A @ A3 )
=> ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_3927_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R2 ) )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A8 )
& ( X2 != Y2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ).
% cofinal_def
thf(fact_3928_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B2: set @ A,I4: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite @ A @ B2 )
=> ( ( 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 @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ 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 @ ( A3 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( 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 ) @ A3 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_3929_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ~ ? [F: A > B] :
( ( inj_on @ A @ B @ F @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A3 ) @ B2 ) ) )
= ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_3930_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( 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 @ B7 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_3931_card__of__Times2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_3932_card__of__Times1,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : A3 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_3933_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 ) @ A3
@ ( product_Sigma @ B @ C @ B2
@ ^ [Uu: B] : C2 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A3 @ B2 )
@ ^ [Uu: A > B] : ( bNF_Wellorder_Func @ A @ C @ A3 @ C2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_3934_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3
@ ^ [Uu: A] : B2 )
@ C2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A3 @ ( bNF_Wellorder_Func @ B @ C @ B2 @ C2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_3935_card__of__Times3,axiom,
! [A: $tType,A3: 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 @ A3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_3936_card__of__Times__same__infinite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite @ A @ A3 )
=> ( 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 @ A3
@ ^ [Uu: A] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_3937_card__of__Pow,axiom,
! [A: $tType,A3: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A3 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_3938_card__of__Pow__Func,axiom,
! [A: $tType,A3: 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 ) @ ( pow @ A @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A3 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_3939_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: 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 @ A3
@ ^ [Uu: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : A3 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_3940_infinite__iff__card__of__nat,axiom,
! [A: $tType,A3: set @ A] :
( ( ~ ( finite_finite @ A @ A3 ) )
= ( 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 @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_3941_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( 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_3942_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A3: 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 ) @ A3 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I4 @ A3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_3943_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A3: A > ( set @ B ),B2: A > ( set @ C )] :
( ! [X3: A] :
( ( member @ A @ X3 @ 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 @ ( A3 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B2 @ X3 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( 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 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I4 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_3944_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ A3
@ ^ [Uu: A] : C2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B2
@ ^ [Uu: B] : C2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_3945_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ C2
@ ^ [Uu: C] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C2
@ ^ [Uu: C] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_3946_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B2: set @ A,I4: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite @ A @ B2 )
=> ( ( 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 @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ 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 @ ( A3 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( 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 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_3947_card__of__refl,axiom,
! [A: $tType,A3: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_3948_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : A3 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_3949_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_3950_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( 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 @ B2
@ ^ [Uu: B] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_3951_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_3952_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( B2
!= ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( 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 @ B2
@ ^ [Uu: B] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_3953_card__of__image,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: 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 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_3954_card__of__empty3,axiom,
! [B: $tType,A: $tType,A3: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_3955_card__of__empty,axiom,
! [B: $tType,A: $tType,A3: 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 @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_3956_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite @ B @ B2 )
=> ( finite_finite @ A @ A3 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_3957_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite @ A @ A3 )
=> ~ ( finite_finite @ B @ B2 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_3958_card__of__mono1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_3959_card__of__empty2,axiom,
! [B: $tType,A: $tType,A3: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_3960_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_3961_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite @ A @ A3 )
= ( finite_finite @ B @ B2 ) ) ) ).
% card_of_ordIso_finite
thf(fact_3962_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_3963_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_3964_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( member @ B @ ( F2 @ A6 ) @ B2 ) )
=> ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_3965_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A3: 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F4: B > A,B9: set @ B] : ( bij_betw @ B @ A @ F4 @ B9 @ A3 ) ) ).
% ex_bij_betw
thf(fact_3966_card__of__least,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A3 @ 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_3967_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_3968_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B2: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ B2 )
=> ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_3969_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( ? [F: A > B] : ( bij_betw @ A @ B @ F @ A3 @ B2 ) )
= ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_3970_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] :
( ! [Y3: 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 @ Y3 ) ) @ 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_3971_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G: B > A] :
( ( image2 @ B @ A @ G @ B2 )
= A3 ) )
= ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_3972_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B2: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_3973_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_3974_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B2: set @ A,A3: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F: B > A] :
( ( image2 @ B @ A @ F @ A3 )
= B2 ) )
= ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_3975_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A3: set @ A,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ C2 )
& ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( 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 @ B7 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_3976_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_3977_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B2: 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 ) ) @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) @ B2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_3978_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C2: 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] : C2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R5 )
@ ^ [Uu: B] : C2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_3979_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),A3: 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 @ A3
@ ^ [Uu: C] : ( field2 @ A @ R3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A3
@ ^ [Uu: C] : ( field2 @ B @ R5 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_3980_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( ? [F: A > B] :
( ( inj_on @ A @ B @ F @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A3 ) @ B2 ) ) )
= ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_3981_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [K7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K7 @ ( field2 @ A @ R2 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K7 @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K7 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_3982_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X: C,X6: 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 )
@ X6 ) ) ) )
@ 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 ) @ X6 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_3983_card__of__ordIso__subst,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_3984_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 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( 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 @ X2 )
@ ^ [Uu: A] : ( order_underS @ A @ R3 @ X2 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_3985_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_3986_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_3987_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_3988_card__order__on__ordIso,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A3 @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A3 @ 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_3989_infinite__Card__order__limit,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( A4 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ R3 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_3990_dir__image,axiom,
! [B: $tType,A: $tType,F2: A > B,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
= ( X3 = Y3 ) )
=> ( ( 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_3991_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_3992_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_3993_card__of__unique,axiom,
! [A: $tType,A3: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A3 @ 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 @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_3994_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R2 )
& ! [R8: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R8 )
=> ( 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 ) ) @ R2 @ R8 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_3995_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( A3
!= ( 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 @ A3
@ ^ [Uu: B] : ( field2 @ A @ R3 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_3996_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( B2
!= ( 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] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_3997_Card__order__Times__same__infinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite @ 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_3998_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 ) @ ( pow @ A @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_3999_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_4000_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_4001_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_4002_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A3: 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 @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_4003_exists__minim__Card__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R )
& ! [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 ) ) @ X3 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_4004_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_4005_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A3: 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 @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite @ A @ ( field2 @ A @ R3 ) )
= ( finite_finite @ B @ A3 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_4006_card__of__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R3 @ A4 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_4007_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As9: A > ( set @ B ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R3 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R3 @ As9 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As9 @ ( 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 @ B2 ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As9 @ X3 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_4008_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B3: 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 @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_4009_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ B] :
( ~ ( finite_finite @ 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 @ A3 ) @ 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 @ B2 ) @ 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 ) @ A3 @ B2 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_4010_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_4011_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ B @ P5 )
!= ( 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 ) ) @ P5 @ 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 @ P5 ) ) )
@ 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 @ P5 )
@ ^ [Uu: B] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_4012_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite @ 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 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ 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 @ ( A3 @ X3 ) ) @ 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 @ A3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_4013_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ C] :
( ~ ( finite_finite @ 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 @ A3 ) @ 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 @ B2 ) @ 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 @ A3
@ ^ [Uu: B] : B2 ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_4014_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite @ 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 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ 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 @ ( A3 @ X3 ) ) @ 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 ) @ A3 @ I4 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_4015_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A3: 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ? [X_1: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A3 @ R3 @ X_1 ) ) ) ).
% ex_toCard_pred
thf(fact_4016_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As9: ( set @ A ) > ( set @ B ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As9 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As9 @ ( 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 @ B2 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As9 @ X3 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_4017_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R8 ) @ R8 )
& ( 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 ) ) ) @ R2 @ R8 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R10: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R10 ) @ R10 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ R10 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R8 @ R10 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_4018_toCard__inj,axiom,
! [B: $tType,A: $tType,A3: 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A3 @ R3 @ X )
= ( bNF_Greatest_toCard @ A @ B @ A3 @ R3 @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% toCard_inj
thf(fact_4019_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_4020_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A3 @ R3 @ ( bNF_Greatest_toCard @ A @ B @ A3 @ R3 ) ) ) ) ).
% toCard_pred_toCard
thf(fact_4021_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_4022_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_4023_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_4024_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_4025_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_4026_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_4027_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_4028_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A3: set @ A,R3: set @ ( product_prod @ B @ B ),B3: 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 @ A3 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ A @ B3 @ A3 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A3 @ R3 @ ( bNF_Greatest_toCard @ A @ B @ A3 @ R3 @ B3 ) )
= B3 ) ) ) ) ).
% fromCard_toCard
thf(fact_4029_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As9: ( set @ A ) > ( set @ B ),B2: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As9 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As9 @ ( 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 @ B2 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As9 @ X3 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_4030_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 )
=> ( ! [X3: 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 @ X3 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X3: 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 @ X3 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( 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_4031_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set @ A,B2: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) )
=> ( ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ A3 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_4032_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 ) @ A3
@ ^ [Uu: A] : ( sum_Plus @ B @ C @ B2 @ C2 ) ) )
@ ( 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 @ A3
@ ^ [Uu: A] : B2 )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu: A] : C2 ) ) ) )
@ ( 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_4033_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 @ B2 ) @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A3 @ ( sum_Plus @ B @ C @ B2 @ C2 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_4034_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: 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 @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B2 @ A3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_4035_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_4036_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B2: set @ A,A3: 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_4037_card__of__Plus__Times__bool,axiom,
! [A: $tType,A3: 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 @ A3 @ A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A3
@ ^ [Uu: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_4038_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A3 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_4039_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_4040_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A3: set @ A,B2: 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 ) @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_4041_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A3: 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 @ A3 @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_4042_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B2: 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 ) @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_4043_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
=> ( bNF_Ca4139267488887388095finite @ B @ R22 ) ) ) ).
% cinfinite_mono
thf(fact_4044_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_4045_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),A3: 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 @ A3 @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A3 @ ( field2 @ B @ R5 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_4046_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C2: 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 ) @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R5 ) @ C2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_4047_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: 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 ) ) @ P5 @ P8 ) @ ( 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 @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R5 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_4048_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),A3: 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 @ A3 @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A3 @ ( field2 @ B @ R5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_4049_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C2: 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 ) @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R5 ) @ C2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_4050_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: 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 ) ) @ P5 @ P8 ) @ ( 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 @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R5 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_4051_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ C2 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C2 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_4052_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ A3 @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B2 @ C2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_4053_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ C2 ) @ ( 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 @ A3 @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B2 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_4054_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ C2 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C2 @ B2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_4055_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ A3 @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B2 @ C2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_4056_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C2: 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( 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 @ C2 ) @ ( 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 @ A3 @ C2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B2 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_4057_cprod__com,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( 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_4058_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 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( X1 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X3 ) @ R3 )
& ( X22 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X3 ) @ R3 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_4059_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 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( X != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X3 ) @ R3 ) ) ) ) ).
% Cinfinite_limit
thf(fact_4060_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P5: 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 ) ) @ P5 @ 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 @ P5 ) @ P5 )
=> ( ( 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 @ P5 @ Q4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_4061_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: 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 @ P5 @ P8 ) @ ( 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 @ P5 @ P8 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_4062_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( 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 @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( 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 @ B2 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_4063_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( 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 @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_4064_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite @ A @ A3 )
=> ( ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( 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 @ B2 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_4065_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C2: set @ A,A3: set @ B,B2: set @ C] :
( ~ ( finite_finite @ A @ C2 )
=> ( ( 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 @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C2 ) ) @ ( 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 @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C2 ) ) @ ( 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 @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C2 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_4066_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) ) ) ) ).
% Cinfinite_cong
thf(fact_4067_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite @ 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 ) ) @ P5 @ 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 @ P5 ) ) ) @ 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 @ P5 ) @ ( field2 @ A @ R3 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_4068_Cinfinite__limit__finite,axiom,
! [A: $tType,X6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ A @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ X6 )
=> ( ( Xa2 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_4069_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_4070_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: 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 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( 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 @ R1 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_mono1
thf(fact_4071_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_4072_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set @ A,B13: B,B23: B,B2: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) )
=> ( ( ( B13 != B23 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B13 @ ( insert2 @ B @ B23 @ ( bot_bot @ ( set @ B ) ) ) ) @ B2 ) )
=> ( 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 @ A3 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_4073_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_4074_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ P22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_4075_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_4076_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ C] :
( ~ ( finite_finite @ 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 @ A3 ) @ 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 @ B2 ) @ 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 @ A3 @ B2 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_4077_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ C] :
( ~ ( finite_finite @ 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 @ A3 ) @ 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 @ B2 ) @ 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 @ A3 @ B2 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_4078_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A3: set @ A,R3: set @ ( product_prod @ B @ B ),B2: 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 @ A3 ) @ 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 @ B2 ) @ 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 ) @ A3 @ B2 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_4079_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set @ A,R3: set @ ( product_prod @ B @ B ),A3: 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 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ 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 @ ( A3 @ X3 ) ) @ 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 ) @ A3 @ I4 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_4080_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I4: set @ B,A3: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ 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 @ ( A3 @ X3 ) ) @ 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 )
@ ^ [I2: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ I2 ) ) )
@ ( 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_4081_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A3: A > ( set @ B ),B2: set @ C] :
( ! [X3: A] :
( ( member @ A @ X3 @ 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 @ ( A3 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) ) @ ( 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 )
@ ^ [I2: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A3 @ I2 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_4082_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( sum_Plus @ A @ B @ A3 @ B2 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_4083_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S2 ) @ S2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R3 @ S2 ) @ S2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_4084_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S2 ) @ S2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ S2 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_4085_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P5: 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 ) ) @ P5 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P5 ) @ P5 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ 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 @ P5 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_4086_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( 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_4087_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) )
& ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ) ).
% Cinfinite_cprod2
thf(fact_4088_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ).
% cinfinite_cprod2
thf(fact_4089_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_4090_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_4091_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_4092_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_4093_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_4094_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_4095_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_4096_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_4097_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_4098_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_4099_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_4100_cexp__mono,axiom,
! [E: $tType,F3: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( 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 @ F3 @ F3 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F3 @ F3 ) ) @ R22 @ ( bNF_Cardinal_czero @ F3 ) ) @ ( bNF_Wellorder_ordIso @ D @ F3 ) ) )
=> ( ( 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 @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_4101_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),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 @ R22 ) @ ( 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 ) ) @ R22 @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_4102_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_4103_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ C @ C ),R32: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( 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 @ R1 @ R22 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R1 @ ( bNF_Cardinal_cprod @ C @ B @ R22 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% cexp_cprod
thf(fact_4104_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_4105_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_4106_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ B @ B ),S2: set @ ( product_prod @ C @ C ),T2: 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 @ S2 ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R3 @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S2 @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_4107_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( 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 @ R22 ) @ ( 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 @ R1 @ R22 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% cexp_cprod_ordLeq
thf(fact_4108_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P22 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R22 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( 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 @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_4109_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: 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 @ R1 ) @ ( 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 @ R1 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_mono1
thf(fact_4110_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( ( field2 @ A @ P22 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R22 )
= ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_4111_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_4112_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R22 ) @ R22 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( 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 @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_4113_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: 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 @ R1 ) @ ( 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 @ R1 @ Q4 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_cong1
thf(fact_4114_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_4115_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_4116_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S2: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S2 ) @ S2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( 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 @ S2 @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_4117_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_4118_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_4119_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_4120_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_4121_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_4122_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_4123_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_4124_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: 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 @ P5 @ P8 ) @ ( 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 @ P5 @ P8 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% csum_dup
thf(fact_4125_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
& ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) ) ) ).
% csum_Cnotzero1
thf(fact_4126_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P5: 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 ) ) @ P5 @ 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 @ P5 ) @ P5 )
=> ( ( 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 @ P5 @ Q4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% csum_cinfinite_bound
thf(fact_4127_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_4128_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cprod @ A @ C @ R1 @ R32 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R1 @ ( bNF_Cardinal_csum @ B @ C @ R22 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% cprod_csum_distrib1
thf(fact_4129_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B ),P32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( 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_4130_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_csum @ C @ D @ R32 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R1 @ R32 ) @ ( bNF_Cardinal_csum @ B @ D @ R22 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% csum_csum
thf(fact_4131_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_4132_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $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 @ R1 @ R22 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% cprod_csum_cexp
thf(fact_4133_csum__com,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( 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_4134_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( bNF_Cardinal_cfinite @ B @ S2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S2 ) @ S2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( 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 @ S2 ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R3 @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
thf(fact_4135_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_4136_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
= ( ^ [A8: 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 @ A8 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_4137_Un__csum,axiom,
! [A: $tType,A3: set @ A,B2: 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 ) @ A3 @ B2 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_4138_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P1 ) @ P1 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( 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_4139_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),P1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( 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_4140_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_cong2
thf(fact_4141_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: 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 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( 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 @ R1 @ Q4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_cong1
thf(fact_4142_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_cong
thf(fact_4143_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: 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 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( 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 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_mono2
thf(fact_4144_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: 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 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( 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 @ R1 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_mono1
thf(fact_4145_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_mono
thf(fact_4146_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_4147_Plus__csum,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: 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 @ A3 @ B2 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% Plus_csum
thf(fact_4148_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
!= ( 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 @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_4149_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),S2: set @ ( product_prod @ C @ C )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( 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 @ S2 ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R3 @ S2 ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% cprod_cexp_csum_cexp_Cinfinite
thf(fact_4150_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ A @ R1 @ R22 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% csum_absorb2'
thf(fact_4151_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% csum_absorb1'
thf(fact_4152_csum__absorb1,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R22 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% csum_absorb1
thf(fact_4153_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_4154_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ B @ B ),F: A > B] :
( ( bij_betw @ A @ B @ F @ ( field2 @ A @ R2 ) @ ( field2 @ B @ R8 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R2 ) )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) @ R8 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_4155_prod__mset__def,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( comm_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_mset_def
thf(fact_4156_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_4157_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_4158_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,C3: A,A4: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ B3 @ C3 )
=> ( Less @ ( F2 @ A4 @ B3 ) @ C3 ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_4159_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,C3: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A4 @ C3 )
=> ( Less @ ( F2 @ A4 @ B3 ) @ C3 ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_4160_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
= ( ( A4
= ( F2 @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_4161_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A4 @ ( F2 @ B3 @ C3 ) )
=> ~ ( ( Less @ A4 @ B3 )
=> ~ ( Less @ A4 @ C3 ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_4162_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,C3: A,A4: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B3 @ C3 )
=> ( Less_eq @ ( F2 @ A4 @ B3 ) @ C3 ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_4163_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,C3: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ C3 )
=> ( Less_eq @ ( F2 @ A4 @ B3 ) @ C3 ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_4164_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ ( F2 @ B3 @ C3 ) )
= ( ( Less_eq @ A4 @ B3 )
& ( Less_eq @ A4 @ C3 ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_4165_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,A4: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B3 @ A4 )
= ( ( F2 @ A4 @ B3 )
= B3 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_4166_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
= ( ( F2 @ A4 @ B3 )
= A4 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_4167_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( Less_eq @ ( F2 @ A4 @ B3 ) @ B3 ) ) ).
% semilattice_order.cobounded2
thf(fact_4168_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( Less_eq @ ( F2 @ A4 @ B3 ) @ A4 ) ) ).
% semilattice_order.cobounded1
thf(fact_4169_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
= ( A4
= ( F2 @ A4 @ B3 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_4170_semilattice__order_OboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less_eq @ A4 @ C3 )
=> ( Less_eq @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_4171_semilattice__order_OboundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ ( F2 @ B3 @ C3 ) )
=> ~ ( ( Less_eq @ A4 @ B3 )
=> ~ ( Less_eq @ A4 @ C3 ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_4172_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,A4: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ B3 @ A4 )
=> ( ( F2 @ A4 @ B3 )
= B3 ) ) ) ).
% semilattice_order.absorb4
thf(fact_4173_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( ( F2 @ A4 @ B3 )
= A4 ) ) ) ).
% semilattice_order.absorb3
thf(fact_4174_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,A4: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B3 @ A4 )
=> ( ( F2 @ A4 @ B3 )
= B3 ) ) ) ).
% semilattice_order.absorb2
thf(fact_4175_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( F2 @ A4 @ B3 )
= A4 ) ) ) ).
% semilattice_order.absorb1
thf(fact_4176_semilattice__order_OorderI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( A4
= ( F2 @ A4 @ B3 ) )
=> ( Less_eq @ A4 @ B3 ) ) ) ).
% semilattice_order.orderI
thf(fact_4177_semilattice__order_OorderE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( A4
= ( F2 @ A4 @ B3 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_4178_semilattice__order_Omono,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,C3: A,B3: A,D3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ C3 )
=> ( ( Less_eq @ B3 @ D3 )
=> ( Less_eq @ ( F2 @ A4 @ B3 ) @ ( F2 @ C3 @ D3 ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_4179_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_4180_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_4181_semilattice__neutr__order__def,axiom,
! [A: $tType] :
( ( semila1105856199041335345_order @ A )
= ( ^ [F: A > A > A,Z3: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_neutr @ A @ F @ Z3 )
& ( semilattice_order @ A @ F @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_neutr_order_def
thf(fact_4182_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_4183_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf.semilattice_order_axioms
thf(fact_4184_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% min.semilattice_order_axioms
thf(fact_4185_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_4186_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_4187_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_4188_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A3: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ S )
=> ( ( finite_finite @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_4189_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,F2: A > A > A,Less: A > A > $o] :
( ! [A6: A,B5: A] :
( ( Less_eq @ A6 @ B5 )
= ( A6
= ( F2 @ A6 @ B5 ) ) )
=> ( ! [A6: A,B5: A] :
( ( Less @ A6 @ B5 )
= ( ( A6
= ( F2 @ A6 @ B5 ) )
& ( A6 != B5 ) ) )
=> ( semila6385135966242565138axioms @ A @ F2 @ Less_eq @ Less ) ) ) ).
% semilattice_order_axioms.intro
thf(fact_4190_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_4191_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_4192_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_4193_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A3: set @ A,X: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_4194_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F5: filter @ A,G6: filter @ B,H9: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F5 @ G6 ) @ 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 )
@ ^ [X2: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y2: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G6 @ H9 ) ) ) ) ).
% prod_filter_assoc
thf(fact_4195_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P3: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P3 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_4196_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P3: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P3 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_4197_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_4198_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_4199_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_4200_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_4201_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 )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ F5 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F2 @ F5 ) @ ( filtermap @ C @ B @ G2 @ F5 ) ) ) ).
% filtermap_Pair
thf(fact_4202_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ! [P4: A > $o] :
( ( eventually @ A @ P4 @ F5 )
=> ? [X4: B] : ( P4 @ ( F2 @ X4 ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_4203_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_4204_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_4205_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ) ).
% trancl_def
thf(fact_4206_reflp__refl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 ) ) ).
% reflp_refl_eq
thf(fact_4207_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 ) )
=> ( ! [A6: A,B5: B] :
( ( R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( ( R3 @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_4208_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_4209_principal__eq__bot__iff,axiom,
! [A: $tType,X6: set @ A] :
( ( ( principal @ A @ X6 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_4210_tranclp__trancl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_4211_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R2: 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 @ R2 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_4212_finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( finite5375528669736107172at_top @ A @ A3 )
= ( principal @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_4213_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I4: set @ A,F5: A > ( set @ B ),F2: B > C,G6: D > ( set @ C ),J4: set @ D] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I4 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I3 ) @ ( F5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J2 ) @ ( F5 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F2
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G6 @ J3 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I2: A] : ( principal @ B @ ( F5 @ I2 ) )
@ I4 ) ) )
= ( ! [X2: D] :
( ( member @ D @ X2 @ J4 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ I4 )
& ! [Z3: B] :
( ( member @ B @ Z3 @ ( F5 @ Y2 ) )
=> ( member @ C @ ( F2 @ Z3 ) @ ( G6 @ X2 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_4214_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X2: B] :
( fChoice @ A
@ ^ [Y2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_4215_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_4216_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert2 @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_4217_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A3: set @ A] :
( ( finite5375528669736107172at_top @ A @ A3 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_4218_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G6: filter @ B,F5: filter @ A,G2: A > C,H9: filter @ C] :
( ( filterlim @ A @ B @ F2 @ G6 @ F5 )
=> ( ( filterlim @ A @ C @ G2 @ H9 @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ ( prod_filter @ B @ C @ G6 @ H9 )
@ F5 ) ) ) ).
% filterlim_Pair
thf(fact_4219_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
@ S ) ) ).
% some_elem
thf(fact_4220_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_4221_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_4222_small__lazy_H_Ocases,axiom,
! [X: product_prod @ int @ int] :
~ ! [D2: int,I3: int] :
( X
!= ( product_Pair @ int @ int @ D2 @ I3 ) ) ).
% small_lazy'.cases
thf(fact_4223_le__prod__encode__2,axiom,
! [B3: nat,A4: nat] : ( ord_less_eq @ nat @ B3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A4 @ B3 ) ) ) ).
% le_prod_encode_2
thf(fact_4224_le__prod__encode__1,axiom,
! [A4: nat,B3: nat] : ( ord_less_eq @ nat @ A4 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A4 @ B3 ) ) ) ).
% le_prod_encode_1
thf(fact_4225_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B2: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A3 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B2 @ ( basic_rel_prod @ A @ B @ C @ D @ A3 @ B2 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_4226_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A4: A,B3: C,C3: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A4 @ B3 ) @ ( product_Pair @ B @ D @ C3 @ D3 ) )
= ( ( R12 @ A4 @ C3 )
& ( R23 @ B3 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_4227_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A4: A,B3: B,R23: C > D > $o,C3: C,D3: D] :
( ( R12 @ A4 @ B3 )
=> ( ( R23 @ C3 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A4 @ C3 ) @ ( product_Pair @ B @ D @ B3 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_4228_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R13: A > B > $o,R24: 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 ) )
& ( R13 @ A5 @ B4 )
& ( R24 @ C5 @ D5 ) ) ) ) ).
% rel_prod.simps
thf(fact_4229_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ A1 @ A22 )
=> ~ ! [A6: A,B5: B,C4: C] :
( ( A1
= ( product_Pair @ A @ C @ A6 @ C4 ) )
=> ! [D2: D] :
( ( A22
= ( product_Pair @ B @ D @ B5 @ D2 ) )
=> ( ( R12 @ A6 @ B5 )
=> ~ ( R23 @ C4 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_4230_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_4231_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P3: A > $o,X2: A] :
! [Y2: A] :
( ( member @ A @ Y2 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P3 @ Y2 ) ) ) ) ).
% ID.pred_set
thf(fact_4232_wfP__SUP,axiom,
! [B: $tType,A: $tType,R3: A > B > B > $o] :
( ! [I3: A] : ( wfP @ B @ ( R3 @ I3 ) )
=> ( ! [I3: A,J2: A] :
( ( ( R3 @ I3 )
!= ( R3 @ J2 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R3 @ I3 ) ) @ ( 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_4233_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list @ A,Xs2: 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 @ Xs2 ) @ 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 @ Xs2 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% mergesort_by_rel_split_length
thf(fact_4234_inf1I,axiom,
! [A: $tType,A3: A > $o,X: A,B2: A > $o] :
( ( A3 @ X )
=> ( ( B2 @ X )
=> ( inf_inf @ ( A > $o ) @ A3 @ B2 @ X ) ) ) ).
% inf1I
thf(fact_4235_inf1E,axiom,
! [A: $tType,A3: A > $o,B2: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B2 @ X )
=> ~ ( ( A3 @ X )
=> ~ ( B2 @ X ) ) ) ).
% inf1E
thf(fact_4236_inf1D1,axiom,
! [A: $tType,A3: A > $o,B2: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B2 @ X )
=> ( A3 @ X ) ) ).
% inf1D1
thf(fact_4237_inf1D2,axiom,
! [A: $tType,A3: A > $o,B2: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B2 @ X )
=> ( B2 @ X ) ) ).
% inf1D2
thf(fact_4238_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_4239_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Top: A] :
( ! [A6: A] : ( Less_eq @ A6 @ Top )
=> ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% ordering_top_axioms.intro
thf(fact_4240_ID_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: A,Pa: A > $o] :
( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( bNF_id_bnf @ ( A > $o ) @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_4241_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya2: A,P: A > $o,Pa: A > $o] :
( ( X = Ya2 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( insert2 @ A @ Ya2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
= ( bNF_id_bnf @ ( A > $o ) @ Pa @ Ya2 ) ) ) ) ).
% ID.pred_cong
thf(fact_4242_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_4243_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 ) ) ) ) ) ).
% Domain_def
thf(fact_4244_wfP__wf__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( wf @ A @ R3 ) ) ).
% wfP_wf_eq
thf(fact_4245_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),N: nat,M2: nat] :
( ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N4 ) ) @ ( nth @ A @ Xs @ N4 ) ) @ R ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M2 ) ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% nth_step_trancl
thf(fact_4246_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list @ A,Xs2: 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 @ Xs2 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs2 ) @ Xs ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs1 ) ) @ ( mset @ A @ Xs2 ) ) ) ).
% mset_mergesort_by_rel_split
thf(fact_4247_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_4248_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_4249_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_4250_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list @ A,Xs2: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs2 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs2 ) ) ).
% mergesort_by_rel_split.simps(1)
thf(fact_4251_lenlex__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( 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 ) @ Xs @ Xs ) @ ( lenlex @ A @ R3 ) ) ) ).
% lenlex_irreflexive
thf(fact_4252_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_4253_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_4254_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( times_times @ A @ A5 ) @ ( F @ ( nth @ B @ Xs3 @ N2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4255_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F @ ( nth @ B @ Xs3 @ N2 ) ) @ ( power_power @ A @ A5 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4256_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M2 )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_4257_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A4: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A4 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_4258_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 ) )
=> ( ! [R11: A > A > $o,Xs4: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R11 @ Xs4 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R11 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs4 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R11 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs4 ) ) ) )
=> ( P @ R11 @ Xs4 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_4259_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R6: A > A > $o,Xs3: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs3 @ ( merges9089515139780605204_merge @ A @ R6 @ ( mergesort_by_rel @ A @ R6 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) ) ) ) ) ).
% mergesort_by_rel.simps
thf(fact_4260_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_4261_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_4262_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R11: A > A > $o,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R11 @ Xs4 ) ) ).
% mergesort_by_rel.cases
thf(fact_4263_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_4264_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_4265_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,Xs22: list @ A] : ( merges9089515139780605204_merge @ A @ R @ ( mergesort_by_rel @ A @ R @ Xs12 ) @ ( mergesort_by_rel @ A @ R @ Xs22 ) )
@ ( 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_4266_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] :
? [I2: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_4267_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( lex @ A @ R2 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4268_zipf__zip,axiom,
! [A: $tType,B: $tType,L1: list @ A,L22: list @ B] :
( ( ( size_size @ ( list @ A ) @ L1 )
= ( size_size @ ( list @ B ) @ L22 ) )
=> ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L1 @ L22 )
= ( zip @ A @ B @ L1 @ L22 ) ) ) ).
% zipf_zip
thf(fact_4269_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4270_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4271_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_4272_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A4: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A4 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( times_times @ A @ A4 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A4 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_4273_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_4274_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_4275_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 ) )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_4276_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 ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_4277_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( 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_4278_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_4279_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X3: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs5 ) )
=> ! [Y3: B,Ys3: list @ B] :
( ( Ys
= ( cons @ B @ Y3 @ Ys3 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs5 @ Ys3 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_4280_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( 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_4281_zip__same,axiom,
! [A: $tType,A4: A,B3: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A4 @ ( set2 @ A @ Xs ) )
& ( A4 = B3 ) ) ) ).
% zip_same
thf(fact_4282_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4283_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X3: A,Xs4: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_4284_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs4 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Xs4: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_4285_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F4: A > B,A6: A,As3: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As3 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_4286_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs13: list @ A,Xs23: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) @ ( nil @ A ) ) )
=> ( ! [Xs13: list @ A,Xs23: list @ A,X3: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [Xs13: list @ A,Xs23: list @ A,X12: A,X23: A,Xs4: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs4 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_4287_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R11: A > A > $o,X3: A,Xs4: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) )
=> ( ! [R11: A > A > $o,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs4 @ ( nil @ A ) ) ) )
=> ~ ! [R11: A > A > $o,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_4288_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R11: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R11: A > A > $o,Sl: list @ A,X3: A,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs4 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_4289_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P4: A > $o,Yes: list @ A,No: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P4 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P4: A > $o,Yes: list @ A,No: list @ A,X3: A,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P4 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X3 @ Xs4 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_4290_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P4: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P4: A > B > $o,A6: A,As3: list @ A,B5: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As3 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ! [P4: A > B > $o,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P4: A > B > $o,V3: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_4291_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L23: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L23 ) )
=> ( ! [V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X12: A,L12: list @ A,X23: A,L23: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X23 @ L23 ) ) ) ) ) ) ).
% merge.cases
thf(fact_4292_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F4: A > B > C,A6: A,As3: list @ A,B5: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As3 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ! [A6: A > B > C,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A6: A > B > C,V3: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_4293_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F4: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F4: A > B,X3: A,Y3: A,Zs: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ~ ! [A6: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_4294_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P4: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( nil @ A ) ) )
=> ( ! [P4: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P4: A > A > $o,X3: A,Y3: A,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ).
% successively.cases
thf(fact_4295_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P4: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( nil @ A ) ) )
=> ~ ! [P4: A > A > $o,X3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_4296_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list @ A,Xs2: list @ A,X1: A,X22: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs2 ) @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs1 ) @ ( cons @ A @ X22 @ Xs2 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_4297_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_4298_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_4299_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_4300_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ! [Xs13: list @ A,Xs23: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) ) ) )
=> ( ! [Xs13: list @ A,Xs23: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs13 ) @ Xs23 ) ) ) )
=> ~ ! [Xs13: list @ A,Xs23: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ! [X12: A,X23: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs4 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs13 ) @ ( cons @ A @ X23 @ Xs23 ) ) @ Xs4 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_4301_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list @ A,Xs2: list @ A,X: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs2 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs2 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_4302_Cons__lenlex__iff,axiom,
! [A: $tType,M2: 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 @ M2 @ 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 @ M2 @ N ) @ R3 ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_4303_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R2 ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_4304_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4305_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 ) ) )
=> ( ! [X3: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xb
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( ( ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs4 @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
& ( ~ ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ Y3 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 ) ) ) ) )
=> ~ ( 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 @ X3 @ Xs4 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) )
=> ( ( ( 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_4306_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 ) ) ) ) ) ) )
=> ( ! [A6: A,As3: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As3 ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X @ A6 @ B5 ) @ ( zipf @ A @ B @ C @ X @ As3 @ 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 @ A6 @ As3 ) @ ( cons @ B @ B5 @ 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_4307_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 ) )
& ! [X2: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y2: A,Z3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y2 @ Z3 ) @ R3 )
@ X2 ) ) ) ) ).
% listrel_iff_zip
thf(fact_4308_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set @ A] :
( ( set_Cons @ A @ A3 @ ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X2: A] : ( cons @ A @ X2 @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_4309_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_4310_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_4311_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_4312_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_4313_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_4314_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_4315_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Zs2: 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 @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_4316_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_4317_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_4318_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 ) )
=> ~ ! [Y3: B,Ys4: list @ B] :
( ( Xs
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_4319_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 ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_4320_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 ) ) )
| ? [X2: A,Y2: B,Xs3: list @ A,Ys2: list @ B] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y2 @ Ys2 ) )
& ( 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 @ Ys2 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_4321_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 ) ) )
=> ~ ! [X3: A,Y3: B,Xs4: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( 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 ) @ Xs4 @ Ys4 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_4322_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_4323_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_4324_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_4325_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_4326_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_4327_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_4328_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4329_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_4330_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_4331_list__encode_Oelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X3: nat,Xs4: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs4 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs4 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_4332_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 ) )
=> ( ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y = X3 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y3: A,Zs: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) )
=> ~ ( ( 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_4333_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_4334_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4335_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_4336_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4337_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ? [Xys2: list @ A,X2: A,Y2: A,Xs6: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs6 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y2 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_4338_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 ) ) ) ) ) ) ).
% listrel_def
thf(fact_4339_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_4340_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 ) ) ) )
=> ~ ! [X3: nat,Xs4: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs4 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs4 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs4 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_4341_lenlex__append2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( 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_4342_append_Osemigroup__axioms,axiom,
! [A: $tType] : ( semigroup @ ( list @ A ) @ ( append @ A ) ) ).
% append.semigroup_axioms
thf(fact_4343_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs2: 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 @ Zs2 ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R3 ) ) ) ).
% lex_append_leftI
thf(fact_4344_lex__append__left__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( 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 ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_left_iff
thf(fact_4345_lex__append__leftD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( 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 ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_leftD
thf(fact_4346_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_4347_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( 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_4348_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_4349_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_4350_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs4 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_4351_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A4: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A4 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A4 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A4 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A4 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_4352_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: list @ A,Y2: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X2 @ Y2 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_4353_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),N2: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N2 )
& ? [Xys2: list @ A,X2: A,Y2: A,Xs6: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs6 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y2 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_4354_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X2: list @ A,Y2: list @ A] :
? [A5: A,V2: list @ A] :
( ( Y2
= ( append @ A @ X2 @ ( cons @ A @ A5 @ V2 ) ) )
| ? [U2: list @ A,B4: A,C5: A,W3: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R2 )
& ( X2
= ( append @ A @ U2 @ ( cons @ A @ B4 @ W3 ) ) )
& ( Y2
= ( append @ A @ U2 @ ( cons @ A @ C5 @ Z3 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_4355_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
? [Us2: list @ A,Z3: A,Z10: A,Vs2: list @ A] :
( ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ Z3 @ Vs2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Z10 ) @ R2 )
& ( Ys2
= ( append @ A @ Us2 @ ( cons @ A @ Z10 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_4356_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_4357_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_4358_lexord__cons__cons,axiom,
! [A: $tType,A4: A,X: list @ A,B3: 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 @ A4 @ X ) @ ( cons @ A @ B3 @ Y ) ) @ ( lexord @ A @ R3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
| ( ( A4 = B3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_4359_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,X2: list @ A] :
( Y
= ( cons @ A @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_4360_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: 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 @ Zs2 ) ) @ ( lexord @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_4361_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_4362_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_4363_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_4364_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_4365_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_4366_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_4367_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_4368_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_4369_lexord__linear,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 )
| ( A6 = B5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ 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_4370_lexord__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( 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 ) @ Xs @ Xs ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_irreflexive
thf(fact_4371_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_4372_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_4373_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_4374_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_4375_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_4376_lexord__asymmetric,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: list @ A,B3: list @ A] :
( ( asym @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A4 @ B3 ) @ ( lexord @ A @ R ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B3 @ A4 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_asymmetric
thf(fact_4377_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_4378_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 ) )
=> ( ! [Y3: A] :
( ( Ys
= ( cons @ A @ Y3 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 ) )
=> ~ ! [Zs: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_4379_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 ) )
=> ( ! [X3: A] :
( ( Xs
= ( cons @ A @ X3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R3 ) )
=> ~ ! [Zs: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_4380_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs2: list @ A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ 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 @ Zs2 ) @ ( lexord @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_4381_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 ) )
=> ( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_append_leftD
thf(fact_4382_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ? [B10: A,Z6: list @ A] :
( Y
= ( cons @ A @ B10 @ Z6 ) )
=> ( 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_4383_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs2: 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 @ Zs2 ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R3 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( 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_4384_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_4385_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_4386_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_4387_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 ) )
=> ~ ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y3 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_4388_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_4389_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_4390_lexord__append__left__rightI,axiom,
! [A: $tType,A4: A,B3: 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 @ A4 @ B3 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A4 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B3 @ Y ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_4391_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: 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 @ Zs2 ) ) @ ( lexord @ A @ R3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( 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 ) @ Ys @ Zs2 ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_4392_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_4393_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_4394_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R2: A > A > $o,Xs3: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_4395_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R2: A > A > $o,Xs3: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_4396_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 ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I2 @ X )
= ( take @ A @ I2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I2 ) @ ( nth @ A @ Y @ I2 ) ) @ R3 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_4397_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_4398_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_4399_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_4400_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_4401_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_4402_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_4403_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 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R3 ) ) ) ) ) ).
% lex_take_index
thf(fact_4404_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_4405_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y3 ) )
=> ( ( 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_4406_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_4407_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4408_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 ) ) ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs4 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_4409_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 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs4 @ ( nil @ A ) ) )
=> ( P @ Xs4 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs4: list @ A,Y3: A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ( P @ Xs4 @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( P @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_4410_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_4411_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_4412_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_4413_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 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Yes2 ) )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ No2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_4414_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_4415_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_4416_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_4417_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 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [X3: A,Xs4: list @ A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 ) )
=> ( ( P @ Ys4 @ Xs4 )
=> ( P @ ( cons @ A @ X3 @ Xs4 ) @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_4418_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X2: A,Uu: list @ A] : ( insert2 @ A @ X2 ) ) ) ).
% set_rec
thf(fact_4419_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 ) ) ) ) )
=> ~ ! [A6: A,As3: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As3 ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As3 @ ( cons @ B @ ( X @ A6 ) @ 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 @ A6 @ As3 ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_4420_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_4421_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_4422_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: 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 ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs2 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_4423_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_4424_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 ) ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( Y
= ( cons @ A @ X3 @ ( splice @ A @ Xa @ Xs4 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_4425_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_4426_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_4427_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_4428_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_4429_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A4: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A4 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A4 ) ) ) ) ).
% distinct_list_update
thf(fact_4430_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 ) )
= ( ? [Y2: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y2 ) @ R3 )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4431_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 )
=> ( ! [X3: A,It: set @ A,Sigma: B] :
( ( member @ A @ X3 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S ) )
=> ( ( I4 @ It @ Sigma )
=> ( I4 @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F2 @ Sigma @ X3 ) ) ) ) )
=> ( I4 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F2 @ Sigma_0 @ S ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_4432_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_4433_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs2 ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y2: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X2: A,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C @ Y2 @ Z3 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs2 ) ) ) ) ).
% zip_left_commute
thf(fact_4434_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 ) )
@ ^ [Y2: B,Ys2: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_4435_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_4436_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_4437_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_4438_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ).
% sum_list_append
thf(fact_4439_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4440_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_4441_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
= L ) ).
% map_snd_mk_fst
thf(fact_4442_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_4443_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R3: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X2: C] : R3
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R3 ) ) ) ).
% sum_list_triv
thf(fact_4444_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G2 @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_4445_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( 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_4446_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
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ 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_4447_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,C3: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C3 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ C3 ) ) ) ).
% sum_list_mult_const
thf(fact_4448_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C3: A,F2: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C3 @ ( F2 @ X2 ) )
@ Xs ) )
= ( times_times @ A @ C3 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4449_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_4450_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
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ 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_4451_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_4452_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
@ ^ [X2: B,Y2: D] : ( F2 @ ( product_Pair @ B @ C @ X2 @ ( G2 @ Y2 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_4453_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
@ ^ [X2: D,Y2: C] : ( F2 @ ( product_Pair @ B @ C @ ( G2 @ X2 ) @ Y2 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_4454_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G2: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G2 @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_4455_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F2: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F2 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_4456_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 )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F2 @ X2 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_4457_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 )
@ ^ [X2: A,Y2: C] : ( product_Pair @ A @ B @ X2 @ ( F2 @ Y2 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_4458_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 )
@ ^ [X2: C,Y2: D] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G2 @ Y2 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ ( map @ D @ B @ G2 @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_4459_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > C > A,G2: A > B > C > B,A4: A,B3: 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,X2: C] : ( product_Pair @ A @ B @ ( F2 @ A5 @ X2 ) @ ( G2 @ A5 @ B4 @ X2 ) ) )
@ ( product_Pair @ A @ B @ A4 @ B3 )
@ Xs ) )
= ( foldl @ A @ C @ F2 @ A4 @ Xs ) ) ).
% fst_foldl
thf(fact_4460_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_4461_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_4462_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Zs2: list @ A] :
( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs2 ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs2 ) ) ) ).
% foldl_absorb1
thf(fact_4463_foldl__un__empty__eq,axiom,
! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
= ( 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_4464_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_4465_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_4466_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_4467_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_4468_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys2: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X2: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X2 ) )
@ ( zip @ B @ A @ Ys2 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_4469_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
@ ^ [X2: A] : X2
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_4470_foldl__set,axiom,
! [A: $tType,L: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
= ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [X2: set @ A] : ( member @ ( set @ A ) @ X2 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_4471_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( 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_4472_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_4473_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_4474_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_4475_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 ) )
@ ^ [Z3: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z3 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_4476_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( 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_4477_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( map @ B @ ( product_prod @ B @ A )
@ ^ [X2: B] : ( product_Pair @ B @ A @ X2 @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_4478_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_fst_mk_fst
thf(fact_4479_prod_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Xs: list @ B,G2: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G2 @ Xs ) ) ) ) ) ).
% prod.distinct_set_conv_list
thf(fact_4480_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_4481_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_4482_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_4483_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_4484_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_4485_interv__sum__list__conv__sum__set__int,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F2: int > B,K: int,L: int] :
( ( groups8242544230860333062m_list @ B @ ( map @ int @ B @ F2 @ ( upto @ K @ L ) ) )
= ( groups7311177749621191930dd_sum @ int @ B @ F2 @ ( set2 @ int @ ( upto @ K @ L ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_int
thf(fact_4486_sum__set__upto__conv__sum__list__int,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: int > A,I: int,J: int] :
( ( groups7311177749621191930dd_sum @ int @ A @ F2 @ ( set2 @ int @ ( upto @ I @ J ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ int @ A @ F2 @ ( upto @ I @ J ) ) ) ) ) ).
% sum_set_upto_conv_sum_list_int
thf(fact_4487_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C3: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C3 ) ) ) ).
% sum_list_replicate
thf(fact_4488_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% prod_list_zero_iff
thf(fact_4489_sum__list__Suc,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X2: A] : ( suc @ ( F2 @ X2 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_4490_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_4491_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_4492_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_4493_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_4494_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_4495_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_4496_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs2 ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( 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 ) ) )
@ ^ [X2: A,Y2: B,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C @ Y2 @ Z3 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs2 ) ) ) ).
% zip_assoc
thf(fact_4497_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X6: set @ A,F2: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X6 )
=> ( ( finite_finite @ A @ X6 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F2 @ X2 ) )
@ X6 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4498_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy2: 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 @ Xy2 )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy2 ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_4499_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
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F2 @ X2 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_4500_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ( 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_4501_length__concat,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ).
% length_concat
thf(fact_4502_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys2: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys2 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_4503_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4504_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [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_4505_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_4506_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 ) ) ) )
=> ~ ! [X3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ Ys4 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X3 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_4507_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 ) ) ) ) )
=> ~ ! [X3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( Y
= ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X3 @ Y2 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_4508_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 ) )
=> ~ ! [X3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ Ys4 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X3 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_4509_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_4510_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_4511_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_4512_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 ) )
=> ! [X3: A,Xs4: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( Y
= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X3 @ No ) ) ) @ Xs4 ) )
=> ~ ( 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 @ X3 @ Xs4 ) ) ) ) ) ) ) ) ) ) ).
% partition_rev.pelims
thf(fact_4513_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_4514_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F @ Xs3 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_4515_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_4516_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_4517_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
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4518_sum__list__filter__le__nat,axiom,
! [A: $tType,F2: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_4519_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F2 @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4520_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
@ ^ [X2: A] : X2 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_4521_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_4522_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_4523_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
@ Zs2 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4524_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
@ Zs2 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4525_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
@ Zs2 )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4526_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
@ Zs2 )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4527_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 ) )
=> ! [X3: A,Xs4: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs4 ) )
=> ( Y
!= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X3 @ No ) ) ) @ Xs4 ) ) ) ) ) ) ).
% partition_rev.elims
thf(fact_4528_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F: B > A,Pivot2: A,Xs3: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F @ X2 ) @ Pivot2 )
@ Xs3 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F @ X2 )
= Pivot2 )
@ Xs3 )
@ ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ Pivot2 @ ( F @ X2 ) )
@ Xs3 ) ) ) ) ) ) ).
% part_def
thf(fact_4529_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 ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs4 ) )
=> ( 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 @ X3 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y2: A] : ( X @ Y2 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_4530_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
@ ^ [Y2: A] : ( R @ Y2 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_4531_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
@ ^ [Y2: A] : ( R @ Y2 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_4532_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 ) ) ) ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( 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 @ X3 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y2: A] : ( X @ Y2 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) ) )
=> ~ ( 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 @ X3 @ Xs4 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_4533_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_4534_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 ) ) )
=> ( ! [R11: 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 ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R11 @ Sl @ ( nil @ A ) ) )
=> ( ! [R11: A > A > $o,Sl: list @ A,X3: A,Xs4: 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 ) ) @ R11 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs4 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y5: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R11 @ Z3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y5 )
= Xa2 )
=> ( P @ R11 @ Sl @ Y5 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y5: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R11 @ Z3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y5 )
= Xa2 )
=> ( P @ R11 @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ R11 @ Sl @ Y5 ) ) @ Xb2 ) ) )
=> ( P @ R11 @ Sl @ ( cons @ A @ X3 @ Xs4 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_4535_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_4536_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F: B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B4: A] : ( plus_plus @ A @ ( F @ X2 ) @ ( times_times @ A @ A5 @ B4 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4537_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A3: A > B > $o,B2: C > D > $o] :
( ( A3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( 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 @ B2 @ A3 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A3 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B2 ) @ A3 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_4538_distinct__n__lists,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_4539_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_4540_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_4541_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_4542_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_4543_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A3: A > B > $o] :
( ( A3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A3 ) @ A3 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_4544_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A3: A > B > $o] :
( ( A3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A3 ) @ A3 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_4545_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_4546_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_4547_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_4548_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 ) ) ) )
=> ~ ! [A6: A,As3: list @ A] :
( ( X
= ( cons @ A @ A6 @ As3 ) )
=> ( ( Y
= ( revg @ A @ As3 @ ( cons @ A @ A6 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A6 @ As3 ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_4549_sum__list__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) )
= ( groups8242544230860333062m_list @ A @ Xs ) ) ) ).
% sum_list_rev
thf(fact_4550_prod__list_Orev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Xs: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( rev @ A @ Xs ) )
= ( groups5270119922927024881d_list @ A @ Xs ) ) ) ).
% prod_list.rev
thf(fact_4551_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_4552_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,Xs: list @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G2 @ ( remdups @ B @ Xs ) ) ) ) ) ).
% prod.set_conv_list
thf(fact_4553_prod__list__def,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_list_def
thf(fact_4554_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,Xs: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G2 @ ( remdups @ B @ Xs ) ) ) ) ) ).
% sum_code
thf(fact_4555_sum__list__def,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_def
thf(fact_4556_monoid__list_OF_Ocong,axiom,
! [A: $tType] :
( ( groups_monoid_F @ A )
= ( groups_monoid_F @ A ) ) ).
% monoid_list.F.cong
thf(fact_4557_monoid__list_OCons,axiom,
! [A: $tType,F2: A > A > A,Z2: A,X: A,Xs: list @ A] :
( ( groups_monoid_list @ A @ F2 @ Z2 )
=> ( ( groups_monoid_F @ A @ F2 @ Z2 @ ( cons @ A @ X @ Xs ) )
= ( F2 @ X @ ( groups_monoid_F @ A @ F2 @ Z2 @ Xs ) ) ) ) ).
% monoid_list.Cons
thf(fact_4558_monoid__list_ONil,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups_monoid_list @ A @ F2 @ Z2 )
=> ( ( groups_monoid_F @ A @ F2 @ Z2 @ ( nil @ A ) )
= Z2 ) ) ).
% monoid_list.Nil
thf(fact_4559_monoid__list_Oappend,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Xs: list @ A,Ys: list @ A] :
( ( groups_monoid_list @ A @ F2 @ Z2 )
=> ( ( groups_monoid_F @ A @ F2 @ Z2 @ ( append @ A @ Xs @ Ys ) )
= ( F2 @ ( groups_monoid_F @ A @ F2 @ Z2 @ Xs ) @ ( groups_monoid_F @ A @ F2 @ Z2 @ Ys ) ) ) ) ).
% monoid_list.append
thf(fact_4560_comm__monoid__list_Orev,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Xs: list @ A] :
( ( groups1828464146339083142d_list @ A @ F2 @ Z2 )
=> ( ( groups_monoid_F @ A @ F2 @ Z2 @ ( rev @ A @ Xs ) )
= ( groups_monoid_F @ A @ F2 @ Z2 @ Xs ) ) ) ).
% comm_monoid_list.rev
thf(fact_4561_monoid__list_Oeq__foldr,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Xs: list @ A] :
( ( groups_monoid_list @ A @ F2 @ Z2 )
=> ( ( groups_monoid_F @ A @ F2 @ Z2 @ Xs )
= ( foldr @ A @ A @ F2 @ Xs @ Z2 ) ) ) ).
% monoid_list.eq_foldr
thf(fact_4562_comm__monoid__list__set_Oset__conv__list,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,G2: B > A,Xs: list @ B] :
( ( groups4802862169904069756st_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( set2 @ B @ Xs ) )
= ( groups_monoid_F @ A @ F2 @ Z2 @ ( map @ B @ A @ G2 @ ( remdups @ B @ Xs ) ) ) ) ) ).
% comm_monoid_list_set.set_conv_list
thf(fact_4563_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,Xs: list @ B,G2: B > A] :
( ( groups4802862169904069756st_set @ A @ F2 @ Z2 )
=> ( ( distinct @ B @ Xs )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( set2 @ B @ Xs ) )
= ( groups_monoid_F @ A @ F2 @ Z2 @ ( map @ B @ A @ G2 @ Xs ) ) ) ) ) ).
% comm_monoid_list_set.distinct_set_conv_list
thf(fact_4564_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 ) ) ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L23: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L23 ) )
=> ( ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ L23 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X23 @ L23 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_4565_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A4: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A4 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q8: nat] : ( product_Pair @ nat @ A @ Q8 @ A4 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_4566_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum_list_upt
thf(fact_4567_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% comm_monoid_mult_class.prod_def
thf(fact_4568_interv__sum__list__conv__sum__set__nat,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F2: nat > B,M2: nat,N: nat] :
( ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ ( upt @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set2 @ nat @ ( upt @ M2 @ N ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_nat
thf(fact_4569_sum__set__upt__conv__sum__list__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set2 @ nat @ ( upt @ M2 @ N ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ nat @ A @ F2 @ ( upt @ M2 @ N ) ) ) ) ) ).
% sum_set_upt_conv_sum_list_nat
thf(fact_4570_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F2: nat > A,M2: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F2 @ ( upt @ N @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F2 @ K4 ) )
@ ( upt @ N @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_4571_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,C2: set @ ( set @ B ),G2: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ( finite_finite @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C2 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C2 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( complete_Sup_Sup @ ( set @ B ) @ C2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F2 @ Z2 ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 ) @ G2 @ C2 ) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
thf(fact_4572_comm__monoid__set_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > A > A,Z2: A,I4: set @ B,A3: B > ( set @ C ),G2: C > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ B @ I4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ( finite_finite @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I4 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F2 @ Z2 @ G2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I4 ) ) )
= ( groups_comm_monoid_F @ A @ B @ F2 @ Z2
@ ^ [X2: B] : ( groups_comm_monoid_F @ A @ C @ F2 @ Z2 @ G2 @ ( A3 @ X2 ) )
@ I4 ) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
thf(fact_4573_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,S: set @ B,A4: B,B3: B > A,C3: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C3 @ K4 ) )
@ S )
= ( F2 @ ( B3 @ A4 ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C3 @ K4 ) )
@ S )
= ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.delta_remove
thf(fact_4574_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,B2: set @ B,G2: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( F2 @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ A3 ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ B2 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
thf(fact_4575_comm__monoid__list__set_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups4802862169904069756st_set @ A @ F2 @ Z2 )
=> ( groups778175481326437816id_set @ A @ F2 @ Z2 ) ) ).
% comm_monoid_list_set.axioms(2)
thf(fact_4576_comm__monoid__set_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( groups778175481326437816id_set @ A @ F2 @ Z2 ) ) ).
% comm_monoid_set.intro
thf(fact_4577_comm__monoid__set_Oaxioms,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( comm_monoid @ A @ F2 @ Z2 ) ) ).
% comm_monoid_set.axioms
thf(fact_4578_comm__monoid__set__def,axiom,
! [A: $tType] :
( ( groups778175481326437816id_set @ A )
= ( comm_monoid @ A ) ) ).
% comm_monoid_set_def
thf(fact_4579_prod_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_set_axioms
thf(fact_4580_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F2: A > A > A,Z2: A,G2: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty
thf(fact_4581_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F2: A > A > A,Z2: A,P5: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_G @ A @ B @ F2 @ Z2 @ P5 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty'
thf(fact_4582_comm__monoid__list__set_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( groups1828464146339083142d_list @ A @ F2 @ Z2 )
=> ( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( groups4802862169904069756st_set @ A @ F2 @ Z2 ) ) ) ).
% comm_monoid_list_set.intro
thf(fact_4583_comm__monoid__list__set__def,axiom,
! [A: $tType] :
( ( groups4802862169904069756st_set @ A )
= ( ^ [F: A > A > A,Z3: A] :
( ( groups1828464146339083142d_list @ A @ F @ Z3 )
& ( groups778175481326437816id_set @ A @ F @ Z3 ) ) ) ) ).
% comm_monoid_list_set_def
thf(fact_4584_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,G2: B > A,X: B] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ B @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( insert2 @ B @ X @ A3 ) )
= ( F2 @ ( G2 @ X ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.insert_remove
thf(fact_4585_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,X: B,G2: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ A3 )
= ( F2 @ ( G2 @ X ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.remove
thf(fact_4586_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ( 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,Xs23: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ~ ( 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 @ Xs23 ) @ ( nil @ A ) ) ) ) ) )
=> ( ! [Xs13: list @ A,Xs23: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs13 ) @ Xs23 ) )
=> ~ ( 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 @ Xs23 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs13: list @ A,Xs23: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs23 ) )
=> ! [X12: A,X23: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs4 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs13 ) @ ( cons @ A @ X23 @ Xs23 ) ) @ Xs4 ) )
=> ~ ( 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 @ Xs23 ) @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs4 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_4587_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_4588_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_4589_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_4590_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A4 = B3 )
= ( ( Less_eq @ A4 @ B3 )
& ( Less_eq @ B3 @ A4 ) ) ) ) ).
% ordering.eq_iff
thf(fact_4591_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less_eq @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% ordering.antisym
thf(fact_4592_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
= ( ( Less @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_4593_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
= ( ( Less_eq @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_4594_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_4595_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A4 != B3 )
=> ( ( Less_eq @ A4 @ B3 )
=> ( Less @ A4 @ B3 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_4596_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B5: A] :
( ( Less_eq @ A6 @ B5 )
= ( ( Less @ A6 @ B5 )
| ( A6 = B5 ) ) )
=> ( ! [A6: A,B5: A] :
( ( Less @ A6 @ B5 )
=> ~ ( Less @ B5 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B5: A,C4: A] :
( ( Less @ A6 @ B5 )
=> ( ( Less @ B5 @ C4 )
=> ( Less @ A6 @ C4 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_4597_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_4598_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_4599_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_4600_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,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A3 ) @ S )
=> ( ( finite_finite @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_4601_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D4 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( 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 ) @ M2 @ X @ Y ) @ D4 )
= ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% restrict_fun_upd
thf(fact_4602_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D4: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_4603_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_4604_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,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( finite_finite @ B @ A3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 )
= ( nil @ B ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4605_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D4: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_4606_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,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A3 ) @ S )
=> ( ( finite_finite @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( insert2 @ B @ X @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_4607_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M2: A > ( option @ B )] :
( ( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( 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 @ M2 @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_4608_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_4609_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X2: A,Y2: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_4610_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_4611_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_4612_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_4613_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X2: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_4614_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_4615_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A3: set @ A,F2: A > B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A3 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X2: A] : ( some @ B @ ( F2 @ X2 ) )
@ A3 ) ) ) ) ) ).
% Some_SUP
thf(fact_4616_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_4617_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_4618_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_4619_top__option__def,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ( top_top @ ( option @ A ) )
= ( some @ A @ ( top_top @ A ) ) ) ) ).
% top_option_def
thf(fact_4620_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A3 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ) ) ) ).
% Some_Sup
thf(fact_4621_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M: A > ( option @ B ),P3: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( ( M @ K4 )
= ( some @ B @ V2 ) )
& ( P3 @ ( product_Pair @ A @ B @ K4 @ V2 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_4622_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ 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 @ M2 )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [V4: B] :
( Uu
= ( product_Pair @ A @ B @ K @ V4 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_4623_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs2: 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 @ Zs2 ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) )
& ( P @ Y )
& ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
& ( P @ X2 ) ) ) ) ).
% extract_Some_iff
thf(fact_4624_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs2: 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 @ Zs2 ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) )
& ( P @ Y )
& ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
& ( P @ X4 ) ) ) ) ).
% extract_SomeE
thf(fact_4625_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_4626_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M2 ) )
= ( M2
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_4627_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M2
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_4628_Sup__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) )
= ( ^ [A8: set @ ( option @ A )] :
( if @ ( option @ A )
@ ( ( A8
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( A8
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
@ ( none @ A )
@ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A8 ) ) ) ) ) ) ) ).
% Sup_option_def
thf(fact_4629_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 ) ) ) )
@ ^ [Y2: 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 ) @ Y2 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4630_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( 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_4631_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( 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_4632_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( 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_4633_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( 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_4634_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X3: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X3 ) ) ) ).
% weak_map_of_SomeI
thf(fact_4635_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_4636_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_4637_sup__option__def,axiom,
! [A: $tType] :
( ( sup @ A )
=> ( ( sup_sup @ ( option @ A ) )
= ( ^ [X2: option @ A,Y2: option @ A] :
( case_option @ ( option @ A ) @ A @ Y2
@ ^ [X9: A] :
( case_option @ ( option @ A ) @ A @ X2
@ ^ [Z3: A] : ( some @ A @ ( sup_sup @ A @ X9 @ Z3 ) )
@ Y2 )
@ X2 ) ) ) ) ).
% sup_option_def
thf(fact_4638_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 ) )
=> ? [Ys4: list @ ( product_prod @ B @ A ),Zs: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys4 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs ) ) )
& ( ( map_of @ B @ A @ Ys4 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_4639_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_4640_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_4641_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ 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 ) ) )
@ T2 )
@ ( F2 @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_4642_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( 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_4643_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B,Y9: 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 @ Y9 ) @ Ys ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_4644_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( 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_4645_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_4646_these__not__empty__eq,axiom,
! [A: $tType,B2: set @ ( option @ A )] :
( ( ( these @ A @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B2
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B2
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_4647_these__empty__eq,axiom,
! [A: $tType,B2: set @ ( option @ A )] :
( ( ( these @ A @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B2
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B2
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_4648_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_4649_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( 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 ) ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( 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 )
@ ^ [Y2: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs4 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y4: A,Z5: A] : Y4 = Z5
@ X3 )
@ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ Xa ) ) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
thf(fact_4650_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),K: A,V: B] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S )
=> ( ( ( 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_4651_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_4652_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_4653_in__graphI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),K: B,V: A] :
( ( ( M2 @ K )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M2 ) ) ) ).
% in_graphI
thf(fact_4654_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M2 ) )
=> ( ( M2 @ K )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_4655_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A3 ) ) )
=> ( member @ A @ K @ A3 ) ) ).
% graph_restrictD(1)
thf(fact_4656_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S )
= ( ^ [X2: A] : ( none @ B ) ) )
= ( S
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_4657_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A3 ) ) )
=> ( ( M2 @ K )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_4658_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A5: A,B4: B] :
( ( Uu
= ( product_Pair @ A @ B @ A5 @ B4 ) )
& ( ( M @ A5 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_4659_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: 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_4660_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y2 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_4661_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_4662_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_4663_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S7: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S7
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S7 ) ) ) ) ).
% dflt_None_set_def
thf(fact_4664_the__dflt__None__set,axiom,
! [A: $tType,X: set @ A] :
( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X ) )
= X ) ).
% the_dflt_None_set
thf(fact_4665_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S7: set @ ( product_prod @ B @ A ),K4: B] :
( eps_Opt @ A
@ ^ [V2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K4 @ V2 ) @ S7 ) ) ) ) ).
% set_to_map_def
thf(fact_4666_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_4667_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( ( dom @ A @ B @ F2 )
= ( bot_bot @ ( set @ A ) ) )
= ( F2
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_4668_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_4669_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( the2 @ B @ ( M @ X2 ) ) )
@ ( dom @ A @ B @ M ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_4670_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 )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( some @ B @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_4671_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M2 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( the2 @ B @ ( M2 @ K4 ) ) )
@ Xs ) )
= M2 ) ) ).
% map_of_map_keys
thf(fact_4672_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z2: A] :
( ( ( M2 @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_4673_graph__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M1 ) @ ( graph @ A @ B @ M22 ) ) ) ) ).
% graph_map_add
thf(fact_4674_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_4675_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_4676_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_4677_ran__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M1 ) @ ( ran @ A @ B @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_4678_map__add__comm,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M1 @ M22 )
= ( map_add @ A @ B @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_4679_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A3: A > ( option @ B ),B2: A > ( option @ B ),C2: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A3 ) @ ( dom @ A @ B @ B2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A3 @ ( map_add @ A @ B @ B2 @ C2 ) )
= ( map_add @ A @ B @ B2 @ ( map_add @ A @ B @ A3 @ C2 ) ) ) ) ).
% map_add_left_comm
thf(fact_4680_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B ),N6: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N6 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M7 ) @ ( dom @ A @ B @ N6 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N6 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_4681_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_4682_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 ) ) ) ) ) ) )
=> ( ! [A6: A,As3: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As3 ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( Y
= ( ( X @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X @ As3 @ 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 @ A6 @ As3 ) @ ( cons @ B @ B5 @ 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_4683_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 ) ) ) ) ) )
=> ~ ! [A6: A,As3: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As3 ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ 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 @ A6 @ As3 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ~ ( ( X @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
thf(fact_4684_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_4685_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_4686_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 ) ) )
=> ( ! [A6: A,As3: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As3 ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ 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 @ A6 @ As3 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ( X @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X @ As3 @ 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_4687_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_4688_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_4689_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
! [A: $tType,H2: multiset @ A] :
( ( bot_bot @ ( set @ ( multiset @ A ) ) )
!= ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ H2 ) ) ).
% subset_mset.not_empty_eq_Iic_eq_empty
thf(fact_4690_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B5: A] :
( ( Less @ A6 @ B5 )
= ( ( Less_eq @ A6 @ B5 )
& ( A6 != B5 ) ) )
=> ( ! [A6: A,B5: A] :
( ( Less_eq @ A6 @ B5 )
=> ( ( Less_eq @ B5 @ A6 )
=> ( A6 = B5 ) ) )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% ordering_axioms.intro
thf(fact_4691_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_4692_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_4693_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_4694_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 )
@ ^ [X2: A,Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_4695_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_4696_preordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B5: A] :
( ( Less_eq @ A6 @ B5 )
= ( ( Less @ A6 @ B5 )
| ( A6 = B5 ) ) )
=> ( ! [A6: A,B5: A] :
( ( Less @ A6 @ B5 )
=> ~ ( Less @ B5 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B5: A,C4: A] :
( ( Less @ A6 @ B5 )
=> ( ( Less @ B5 @ C4 )
=> ( Less @ A6 @ C4 ) ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_4697_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( Less_eq @ A4 @ B3 ) ) ) ).
% preordering.strict_implies_order
thf(fact_4698_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
= ( ( Less_eq @ A4 @ B3 )
& ~ ( Less_eq @ B3 @ A4 ) ) ) ) ).
% preordering.strict_iff_not
thf(fact_4699_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( ( Less_eq @ B3 @ C3 )
=> ( Less @ A4 @ C3 ) ) ) ) ).
% preordering.strict_trans2
thf(fact_4700_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less @ B3 @ C3 )
=> ( Less @ A4 @ C3 ) ) ) ) ).
% preordering.strict_trans1
thf(fact_4701_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,A4: A,B3: A,C3: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less_eq @ B3 @ C3 )
=> ( Less_eq @ A4 @ C3 ) ) ) ) ).
% partial_preordering.trans
thf(fact_4702_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o] :
( ! [A6: A] : ( Less_eq @ A6 @ A6 )
=> ( ! [A6: A,B5: A,C4: A] :
( ( Less_eq @ A6 @ B5 )
=> ( ( Less_eq @ B5 @ C4 )
=> ( Less_eq @ A6 @ C4 ) ) )
=> ( partial_preordering @ A @ Less_eq ) ) ) ).
% partial_preordering.intro
thf(fact_4703_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( ( Less @ B3 @ C3 )
=> ( Less @ A4 @ C3 ) ) ) ) ).
% preordering.strict_trans
thf(fact_4704_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: A > A > $o,A4: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( Less_eq @ A4 @ A4 ) ) ).
% partial_preordering.refl
thf(fact_4705_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ~ ( Less @ A4 @ A4 ) ) ).
% preordering.irrefl
thf(fact_4706_preordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ~ ( Less @ B3 @ A4 ) ) ) ).
% preordering.asym
thf(fact_4707_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_4708_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_4709_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% order.partial_preordering_axioms
thf(fact_4710_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_4711_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A
@ ^ [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_4712_order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.preordering_axioms
thf(fact_4713_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_4714_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_4715_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_4716_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B5: A] :
( ( Less @ A6 @ B5 )
= ( ( Less_eq @ A6 @ B5 )
& ~ ( Less_eq @ B5 @ A6 ) ) )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering_axioms.intro
thf(fact_4717_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_4718_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A8: set @ A,F: ( product_prod @ A @ B ) > C,A5: A] :
( if @ ( B > C ) @ ( member @ A @ A5 @ A8 )
@ ^ [B4: B] : ( F @ ( product_Pair @ A @ B @ A5 @ B4 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_4719_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_4720_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_4721_subset__mset_Onot__empty__eq__Ici__eq__empty,axiom,
! [A: $tType,L: multiset @ A] :
( ( bot_bot @ ( set @ ( multiset @ A ) ) )
!= ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L ) ) ).
% subset_mset.not_empty_eq_Ici_eq_empty
thf(fact_4722_relH__def,axiom,
( relH
= ( ^ [As4: set @ nat,H: heap_ext @ product_unit,H6: heap_ext @ product_unit] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As4 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) )
& ! [T6: typerep,X2: nat] :
( ( member @ nat @ X2 @ As4 )
=> ( ( ( refs @ product_unit @ H @ T6 @ X2 )
= ( refs @ product_unit @ H6 @ T6 @ X2 ) )
& ( ( arrays @ product_unit @ H @ T6 @ X2 )
= ( arrays @ product_unit @ H6 @ T6 @ X2 ) ) ) ) ) ) ) ).
% relH_def
thf(fact_4723_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P3: A > $o,Xs3: 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 ) ) ) )
@ ^ [Y2: 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 @ (~) @ P3 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Ys2 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_4724_in__range_Opelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( Y
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ As3 )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ) ) ).
% in_range.pelims(1)
thf(fact_4725_in__range_Opelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ~ ! [X4: nat] :
( ( member @ nat @ X4 @ As3 )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ).
% in_range.pelims(2)
thf(fact_4726_in__range_Opelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ! [X3: nat] :
( ( member @ nat @ X3 @ As3 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H3 ) ) ) ) ) ) ) ).
% in_range.pelims(3)
thf(fact_4727_one__assn__raw_Opelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( Y
= ( As3
= ( bot_bot @ ( set @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) ) ) ) ) ) ).
% one_assn_raw.pelims(1)
thf(fact_4728_one__assn__raw_Opelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( As3
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
thf(fact_4729_one__assn__raw_Opelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H3: heap_ext @ product_unit,As3: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As3 ) )
=> ( As3
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
thf(fact_4730_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R2: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R2 )
@ ^ [X2: B,Y2: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X2 @ Y2 ) @ S5 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_4731_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y2 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V2 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4732_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R2: A > A > $o,X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_4733_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_4734_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_4735_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 )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( ( R3 @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_4736_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A6: A,B5: B] :
( ( R3 @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_4737_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 )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( R3 @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( 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 @ A6 @ B5 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_4738_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_4739_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 )
@ ^ [X2: B,Y2: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X2 @ Y2 ) @ S2 ) )
= ( ^ [X2: A,Y2: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y2 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_4740_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y2 @ V2 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4741_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 ) )
@ ^ [X2: nat,Y2: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V2 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4742_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 )
@ ^ [X2: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_4743_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_4744_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_4745_accp__acc__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( accp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( acc @ A @ R3 ) ) ) ) ).
% accp_acc_eq
thf(fact_4746_acc__def,axiom,
! [A: $tType] :
( ( acc @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ( accp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R2 ) ) ) ) ) ).
% acc_def
thf(fact_4747_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: code_natural > A,N4: code_natural,M3: code_natural] :
( X
!= ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N4 @ M3 ) ) ) ).
% Lazy_Sequence.iterate_upto.cases
thf(fact_4748_acc__induct__rule,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A4 @ ( acc @ A @ R3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( acc @ A @ R3 ) )
=> ( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A4 ) ) ) ).
% acc_induct_rule
thf(fact_4749_not__acc__down,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ ( acc @ A @ R ) )
=> ~ ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X ) @ R )
=> ( member @ A @ Z4 @ ( acc @ A @ R ) ) ) ) ).
% not_acc_down
thf(fact_4750_acc__downward,axiom,
! [A: $tType,B3: A,R3: set @ ( product_prod @ A @ A ),A4: A] :
( ( member @ A @ B3 @ ( acc @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R3 )
=> ( member @ A @ A4 @ ( acc @ A @ R3 ) ) ) ) ).
% acc_downward
thf(fact_4751_acc__induct,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A4 @ ( acc @ A @ R3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( acc @ A @ R3 ) )
=> ( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A4 ) ) ) ).
% acc_induct
thf(fact_4752_acc_Ointros,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R3 )
=> ( member @ A @ Y3 @ ( acc @ A @ R3 ) ) )
=> ( member @ A @ X @ ( acc @ A @ R3 ) ) ) ).
% acc.intros
thf(fact_4753_acc_Osimps,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A4 @ ( acc @ A @ R3 ) )
= ( ? [X2: A] :
( ( A4 = X2 )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R3 )
=> ( member @ A @ Y2 @ ( acc @ A @ R3 ) ) ) ) ) ) ).
% acc.simps
thf(fact_4754_acc_Ocases,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A4 @ ( acc @ A @ R3 ) )
=> ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ A4 ) @ R3 )
=> ( member @ A @ Y5 @ ( acc @ A @ R3 ) ) ) ) ).
% acc.cases
thf(fact_4755_log_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
~ ! [B5: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ B5 @ I3 ) ) ).
% log.cases
thf(fact_4756_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 )] :
~ ! [F4: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I3: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I3 @ D2 ) ) ) ) ).
% exhaustive_fun'.cases
thf(fact_4757_exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I3 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_4758_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 )] :
~ ! [F4: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I3: 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 ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I3 @ D2 ) ) ) ) ).
% full_exhaustive_fun'.cases
thf(fact_4759_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 )] :
~ ! [F4: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I3: 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 ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I3 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_4760_acc__subset__induct,axiom,
! [A: $tType,D4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ D4 @ ( acc @ A @ R ) )
=> ( ! [X3: A,Z4: A] :
( ( member @ A @ X3 @ D4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X3 ) @ R )
=> ( member @ A @ Z4 @ D4 ) ) )
=> ( ( member @ A @ X @ D4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ D4 )
=> ( ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X3 ) @ R )
=> ( P @ Z6 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_4761_acc__downwards,axiom,
! [A: $tType,A4: A,R3: set @ ( product_prod @ A @ A ),B3: A] :
( ( member @ A @ A4 @ ( acc @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ A @ B3 @ ( acc @ A @ R3 ) ) ) ) ).
% acc_downwards
thf(fact_4762_acc__downwards__aux,axiom,
! [A: $tType,B3: A,A4: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ A @ A4 @ ( acc @ A @ R3 ) )
=> ( member @ A @ B3 @ ( acc @ A @ R3 ) ) ) ) ).
% acc_downwards_aux
thf(fact_4763_split__seed__def,axiom,
( split_seed
= ( ^ [S5: 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 ) )
@ ^ [V4: 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 @ V4 @ ( 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 @ S5 ) ) )
@ S5 ) ) ) ).
% split_seed_def
thf(fact_4764_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 ) ) )
=> ( ! [N4: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) )
=> ( ! [X4: nat,Y5: nat] :
( ( ( product_Pair @ nat @ nat @ X4 @ Y5 )
= ( nat_prod_decode @ N4 ) )
=> ( P @ Y5 ) )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_4765_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_4766_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_4767_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 ) ) )
=> ( ! [F4: code_natural > A,N4: 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 ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N4 @ M3 ) ) )
=> ( ! [X4: product_unit] :
( ~ ( ord_less @ code_natural @ M3 @ N4 )
=> ( P @ F4 @ ( plus_plus @ code_natural @ N4 @ ( one_one @ code_natural ) ) @ M3 ) )
=> ( P @ F4 @ N4 @ M3 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_4768_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_4769_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_4770_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K4: code_natural,F: B > A > ( product_prod @ B @ A ),X2: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K4
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X2 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F @ X2 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K4 @ ( one_one @ code_natural ) ) @ F ) ) ) ) ) ).
% iterate.simps
thf(fact_4771_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_4772_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_4773_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_4774_iter_H_Ocases,axiom,
! [A: $tType] :
( ( quickcheck_random @ A )
=> ! [X: product_prod @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) )] :
~ ! [T3: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T3 @ ( 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_4775_select__weight__def,axiom,
! [A: $tType] :
( ( select_weight @ A )
= ( ^ [Xs3: 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 ) @ Xs3 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K4 ) ) ) ) ) ).
% select_weight_def
thf(fact_4776_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
~ ! [K2: code_natural,F4: B > A > ( product_prod @ B @ A ),X3: 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 @ F4 @ X3 ) ) ) ).
% iterate.cases
thf(fact_4777_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_4778_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_4779_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_4780_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_4781_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_4782_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_4783_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs3: 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 ) @ Xs3 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K4 ) ) ) ) ) ) ).
% select_def
thf(fact_4784_Random__Pred_Ounion__def,axiom,
! [A: $tType] :
( ( random_union @ A )
= ( ^ [R13: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R24: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S5: 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 ) )
@ ^ [P12: pred @ A,S8: 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 ) )
@ ^ [P23: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P12 @ P23 ) )
@ ( R24 @ S8 ) )
@ ( R13 @ S5 ) ) ) ) ).
% Random_Pred.union_def
thf(fact_4785_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 ) ) ) ) ) )
=> ( ! [N4: num] :
( ( X
= ( bit0 @ N4 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N4 ) @ one2 ) ) ) ) )
=> ( ! [N4: num] :
( ( X
= ( bit0 @ N4 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N4 ) @ ( bit0 @ M3 ) ) ) ) ) )
=> ( ! [N4: num] :
( ( X
= ( bit0 @ N4 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N4 ) @ ( bit1 @ M3 ) ) ) ) ) )
=> ( ! [N4: num] :
( ( X
= ( bit1 @ N4 ) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N4 ) @ one2 ) ) ) ) )
=> ( ! [N4: num] :
( ( X
= ( bit1 @ N4 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N4 ) @ ( bit0 @ M3 ) ) ) ) ) )
=> ~ ! [N4: num] :
( ( X
= ( bit1 @ N4 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N4 ) @ ( bit1 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_4786_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_4787_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( random_bind @ A @ B )
= ( ^ [R6: ( 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 ) ),S5: 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 ) )
@ ^ [P3: pred @ A,S8: 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 )
@ ( bind @ A @ B @ P3
@ ^ [A5: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F @ A5 @ S1 ) ) ) )
@ ( split_seed @ S8 ) )
@ ( R6 @ S5 ) ) ) ) ).
% Random_Pred.bind_def
thf(fact_4788_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 )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_4789_bottom__bind,axiom,
! [B: $tType,A: $tType,P: B > ( pred @ A )] :
( ( bind @ B @ A @ ( bot_bot @ ( pred @ B ) ) @ P )
= ( bot_bot @ ( pred @ A ) ) ) ).
% bottom_bind
thf(fact_4790_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 )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N7: num] : ( some @ num @ ( bit1 @ N7 ) )
@ ( bit_and_not_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_4791_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 )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( some @ num @ ( bit1 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( some @ num @ ( bit0 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_4792_singleton__sup,axiom,
! [A: $tType,A3: pred @ A,Default: product_unit > A,B2: pred @ A] :
( ( ( A3
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( singleton @ A @ Default @ B2 ) ) )
& ( ( A3
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( ( B2
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( singleton @ A @ Default @ A3 ) ) )
& ( ( B2
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( ( ( singleton @ A @ Default @ A3 )
= ( singleton @ A @ Default @ B2 ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( singleton @ A @ Default @ A3 ) ) )
& ( ( ( singleton @ A @ Default @ A3 )
!= ( singleton @ A @ Default @ B2 ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( Default @ product_Unity ) ) ) ) ) ) ) ) ).
% singleton_sup
thf(fact_4793_singleton__bot,axiom,
! [A: $tType,Default: product_unit > A] :
( ( singleton @ A @ Default @ ( bot_bot @ ( pred @ A ) ) )
= ( Default @ product_Unity ) ) ).
% singleton_bot
thf(fact_4794_singleton__sup__aux,axiom,
! [A: $tType,A3: pred @ A,Default: product_unit > A,B2: pred @ A] :
( ( ( A3
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( singleton @ A @ Default @ B2 ) ) )
& ( ( A3
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( ( B2
= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( singleton @ A @ Default @ A3 ) ) )
& ( ( B2
!= ( bot_bot @ ( pred @ A ) ) )
=> ( ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ A3 @ B2 ) )
= ( singleton @ A @ Default @ ( sup_sup @ ( pred @ A ) @ ( single @ A @ ( singleton @ A @ Default @ A3 ) ) @ ( single @ A @ ( singleton @ A @ Default @ B2 ) ) ) ) ) ) ) ) ) ).
% singleton_sup_aux
thf(fact_4795_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A5: nat,X2: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P7: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q8: num] : ( some @ num @ ( bit0 @ Q8 ) )
@ ( bit_take_bit_num @ O @ P7 ) )
@ ^ [P7: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P7 ) ) )
@ X2 )
@ A5 )
@ ( product_Pair @ nat @ num @ N2 @ M ) ) ) ) ).
% take_bit_num_code
thf(fact_4796_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 )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( bit1 @ N4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( bit1 @ N4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( bit0 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
=> ( ! [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 ) )
=> ! [N4: num] :
( ( Xa
= ( bit0 @ N4 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N4: num] :
( ( Xa
= ( bit1 @ N4 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
thf(fact_4797_unit__pred__cases,axiom,
! [P: ( pred @ product_unit ) > $o,Q: pred @ product_unit] :
( ( P @ ( bot_bot @ ( pred @ product_unit ) ) )
=> ( ( P @ ( single @ product_unit @ product_Unity ) )
=> ( P @ Q ) ) ) ).
% unit_pred_cases
thf(fact_4798_single__not__bot,axiom,
! [A: $tType,X: A] :
( ( single @ A @ X )
!= ( bot_bot @ ( pred @ A ) ) ) ).
% single_not_bot
thf(fact_4799_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_4800_Random__Pred_Osingle__def,axiom,
! [A: $tType] :
( ( random_single @ A )
= ( ^ [X2: A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ A @ X2 ) ) ) ) ).
% Random_Pred.single_def
thf(fact_4801_pred__of__set__set__foldr__sup,axiom,
! [A: $tType,Xs: list @ A] :
( ( pred_of_set @ A @ ( set2 @ A @ Xs ) )
= ( foldr @ ( pred @ A ) @ ( pred @ A ) @ ( sup_sup @ ( pred @ A ) ) @ ( map @ A @ ( pred @ A ) @ ( single @ A ) @ Xs ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% pred_of_set_set_foldr_sup
thf(fact_4802_pred__of__set__fold__sup,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ( ( pred_of_set @ A @ A3 )
= ( finite_fold @ ( pred @ A ) @ ( pred @ A ) @ ( sup_sup @ ( pred @ A ) ) @ ( bot_bot @ ( pred @ A ) ) @ ( image2 @ A @ ( pred @ A ) @ ( single @ A ) @ A3 ) ) ) ) ).
% pred_of_set_fold_sup
thf(fact_4803_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_4804_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_4805_contains__pred__def,axiom,
! [A: $tType] :
( ( predic7144156976422707464s_pred @ A )
= ( ^ [A8: set @ A,X2: A] : ( if @ ( pred @ product_unit ) @ ( member @ A @ X2 @ A8 ) @ ( single @ product_unit @ product_Unity ) @ ( bot_bot @ ( pred @ product_unit ) ) ) ) ) ).
% contains_pred_def
thf(fact_4806_Random__Pred_Onot__randompred__def,axiom,
( random6974930770145893639ompred
= ( ^ [P3: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ),S5: 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 ) )
@ ^ [P9: pred @ product_unit,S8: product_prod @ code_natural @ code_natural] : ( if @ ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( eval @ product_unit @ P9 @ product_Unity ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ product_unit ) ) @ S8 ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ product_unit @ product_Unity ) @ S8 ) )
@ ( P3 @ S5 ) ) ) ) ).
% Random_Pred.not_randompred_def
thf(fact_4807_eval__bot,axiom,
! [A: $tType] :
( ( eval @ A @ ( bot_bot @ ( pred @ A ) ) )
= ( bot_bot @ ( A > $o ) ) ) ).
% eval_bot
thf(fact_4808_not__bot,axiom,
! [A: $tType,A3: pred @ A] :
( ( A3
!= ( bot_bot @ ( pred @ A ) ) )
=> ~ ! [X3: A] :
~ ( eval @ A @ A3 @ X3 ) ) ).
% not_bot
thf(fact_4809_botE,axiom,
! [A: $tType,X: A] :
~ ( eval @ A @ ( bot_bot @ ( pred @ A ) ) @ X ) ).
% botE
thf(fact_4810_not__pred__eq,axiom,
( not_pred
= ( ^ [P3: pred @ product_unit] : ( if @ ( pred @ product_unit ) @ ( eval @ product_unit @ P3 @ product_Unity ) @ ( bot_bot @ ( pred @ product_unit ) ) @ ( single @ product_unit @ product_Unity ) ) ) ) ).
% not_pred_eq
thf(fact_4811_bot__pred__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( pred @ A ) )
= ( pred2 @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_pred_def
thf(fact_4812_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F2: code_natural > A,N: code_natural,M2: 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 @ M2 ) ) )
=> ( ( iterate_upto @ A @ F2 @ N @ M2 )
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M2 @ N ) @ ( empty @ A ) @ ( insert @ A @ ( F2 @ N ) @ ( iterate_upto @ A @ F2 @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M2 ) ) ) ) ) ) ).
% Predicate.iterate_upto.psimps
thf(fact_4813_bot__set__code,axiom,
! [A: $tType] :
( ( bot_bot @ ( pred @ A ) )
= ( seq2 @ A
@ ^ [U2: product_unit] : ( empty @ A ) ) ) ).
% bot_set_code
thf(fact_4814_Predicate_Osingle__code,axiom,
! [A: $tType] :
( ( single @ A )
= ( ^ [X2: A] :
( seq2 @ A
@ ^ [U2: product_unit] : ( insert @ A @ X2 @ ( bot_bot @ ( pred @ A ) ) ) ) ) ) ).
% Predicate.single_code
thf(fact_4815_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_4816_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType] :
( ( random_iterate_upto @ A )
= ( ^ [F: code_natural > A,N2: code_natural,M: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F @ N2 @ M ) ) ) ) ).
% Random_Pred.iterate_upto_def
thf(fact_4817_of__seq__code_I1_J,axiom,
! [A: $tType] :
( ( set_of_seq @ A @ ( empty @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% of_seq_code(1)
thf(fact_4818_pred__of__seq_Osimps_I1_J,axiom,
! [A: $tType] :
( ( pred_of_seq @ A @ ( empty @ A ) )
= ( bot_bot @ ( pred @ A ) ) ) ).
% pred_of_seq.simps(1)
thf(fact_4819_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 ) )
@ ^ [X2: A,P3: pred @ A] : ( insert2 @ A @ X2 @ ( set_of_pred @ A @ P3 ) )
@ ^ [P3: pred @ A,Xq: seq @ A] : ( sup_sup @ ( set @ A ) @ ( set_of_pred @ A @ P3 ) @ ( set_of_seq @ A @ Xq ) )
@ ( F2 @ product_Unity ) ) ) ).
% of_pred_code
thf(fact_4820_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A3: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( P @ X3 ) )
=> ( P @ ( Lub @ A3 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_4821_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A9: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( P @ X4 ) )
=> ( P @ ( Lub @ A9 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_4822_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P3: A > $o] :
! [A8: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P3 @ X2 ) )
=> ( P3 @ ( Lub2 @ A8 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_4823_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,F2: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( P @ ( F2 @ X3 ) ) )
=> ( P @ ( comple115746919287870866o_fixp @ A @ F2 ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_4824_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),V: B] :
( ( map_mmupd @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) @ V )
= M2 ) ).
% map_mmupd_empty
thf(fact_4825_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_4826_prod__mset_Ocomm__monoid__mset__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid_mset @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_mset.comm_monoid_mset_axioms
thf(fact_4827_execute__change,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F2: A > A,R3: ref @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_change @ A @ F2 @ R3 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F2 @ ( ref_get @ A @ H2 @ R3 ) ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ ( F2 @ ( ref_get @ A @ H2 @ R3 ) ) @ H2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% execute_change
thf(fact_4828_successively_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ( ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) )
=> ~ ! [X3: A,Y3: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) ) )
=> ~ ( ( X @ X3 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ) ) ) ).
% successively.pelims(2)
thf(fact_4829_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_4830_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_4831_comm__monoid__mset__def,axiom,
! [A: $tType] :
( ( comm_monoid_mset @ A )
= ( comm_monoid @ A ) ) ).
% comm_monoid_mset_def
thf(fact_4832_successively_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X3: A,Y3: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) ) )
=> ( ( X @ X3 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ) ).
% successively.pelims(3)
thf(fact_4833_successively_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ( ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X3: A,Y3: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) )
=> ( ( Y
= ( ( X @ X3 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ) ) ) ) ).
% successively.pelims(1)
thf(fact_4834_execute__lookup,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R3 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( ref_get @ A @ H2 @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_lookup
thf(fact_4835_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N4: nat,R4: A,H3: heap_ext @ product_unit,N8: nat] :
( X
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N8 ) ) ) ) )
=> ~ ! [N4: nat] :
( X
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% timeFrame.cases
thf(fact_4836_Heap__cases,axiom,
! [A: $tType,F2: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ! [X3: A,H4: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F2 @ H2 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H4 ) ) )
=> ( ( heap_Time_execute @ A @ F2 @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_4837_execute__tap,axiom,
! [A: $tType,F2: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F2 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F2 @ H2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ).
% execute_tap
thf(fact_4838_execute__ureturn,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ).
% execute_ureturn
thf(fact_4839_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
( ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_4840_execute__return,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).
% execute_return
thf(fact_4841_execute__update,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,V: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R3 @ V ) @ H2 )
= ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ V @ H2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_update
thf(fact_4842_ureturn__def,axiom,
! [A: $tType] :
( ( heap_Time_ureturn @ A )
= ( ^ [X2: A] :
( heap_Time_heap @ A
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ) ).
% ureturn_def
thf(fact_4843_return__def,axiom,
! [A: $tType] :
( ( heap_Time_return @ A )
= ( ^ [X2: A] :
( heap_Time_heap @ A
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% return_def
thf(fact_4844_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_update @ A )
= ( ^ [R2: ref @ A,V2: A] :
( heap_Time_heap @ product_unit
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R2 @ V2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Ref_Time.update_def
thf(fact_4845_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y )
=> ( ! [R4: A,H3: heap_ext @ product_unit,N8: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N8 ) ) ) )
=> ( Y
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ X @ N8 ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( Y
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% timeFrame.elims
thf(fact_4846_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ Xa ) )
=> ( ! [R4: A,H3: heap_ext @ product_unit,N8: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N8 ) ) ) )
=> ( ( Y
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ X @ N8 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N8 ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( Y
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% timeFrame.pelims
thf(fact_4847_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R3: A,H2: heap_ext @ product_unit,N6: nat] :
( ( heap_Time_timeFrame @ A @ N @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N6 ) ) ) )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ N @ N6 ) ) ) ) ) ).
% timeFrame.simps(1)
thf(fact_4848_tap__def,axiom,
! [A: $tType] :
( ( heap_Time_tap @ A )
= ( ^ [F: ( heap_ext @ product_unit ) > A] :
( heap_Time_Heap2 @ A
@ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% tap_def
thf(fact_4849_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H5: heap_ext @ product_unit,N: nat,G2: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F2 @ G2 ) @ H2 )
= ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G2 @ X ) @ H5 ) ) ) ) ).
% execute_bind(1)
thf(fact_4850_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H5: heap_ext @ product_unit,N: nat,G2: A > ( heap_Time_Heap @ B ),Y: B,H10: heap_ext @ product_unit,N6: nat] :
( ( ( heap_Time_execute @ A @ F2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G2 @ X ) @ H5 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H10 @ N6 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F2 @ G2 ) @ H2 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H10 @ ( plus_plus @ nat @ N @ N6 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_4851_wait__def,axiom,
( heap_Time_wait
= ( ^ [N2: nat] :
( heap_Time_Heap2 @ product_unit
@ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N2 ) ) ) ) ) ) ).
% wait_def
thf(fact_4852_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H5: heap_ext @ product_unit,N: nat,G2: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N ) ) ) )
=> ( ( heap_Time_success @ B @ ( G2 @ X ) @ H5 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F2 @ G2 ) @ H2 ) ) ) ).
% success_bind_executeI
thf(fact_4853_execute__nth_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ A @ ( array_nth @ A @ A4 @ I ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_nth(1)
thf(fact_4854_successE,axiom,
! [A: $tType,F2: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F2 @ H2 )
=> ~ ! [R4: A,H4: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F2 @ H2 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ H4 ) ) ) ) ).
% successE
thf(fact_4855_Array__Time_Onth__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_nth @ A )
= ( ^ [A5: array @ A,I2: nat] :
( heap_Time_guard @ A
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H @ A5 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A5 ) @ I2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Array_Time.nth_def
thf(fact_4856_freeze__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_freeze @ A )
= ( ^ [A5: array @ A] :
( heap_Time_heap @ ( list @ A )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H @ A5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H @ A5 ) ) ) ) ) ) ) ) ).
% freeze_def
thf(fact_4857_execute__freeze,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( list @ A ) @ ( array_freeze @ A @ A4 ) @ H2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H2 @ A4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H2 @ A4 ) ) ) ) ) ) ) ).
% execute_freeze
thf(fact_4858_execute__swap_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A4 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A4 @ I @ X @ H2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% execute_swap(1)
thf(fact_4859_swap__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_swap @ A )
= ( ^ [I2: nat,X2: A,A5: array @ A] :
( heap_Time_guard @ A
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H @ A5 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A5 ) @ I2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I2 @ X2 @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% swap_def
thf(fact_4860_execute__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ nat @ ( array_len @ A @ A4 ) @ H2 )
= ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H2 @ A4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_len
thf(fact_4861_map__entry__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_map_entry @ A )
= ( ^ [I2: nat,F: A > A,A5: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H @ A5 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I2 @ ( F @ ( nth @ A @ ( array_get @ A @ H @ A5 ) @ I2 ) ) @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% map_entry_def
thf(fact_4862_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,F2: A > A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F2 @ A4 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A4 @ I @ ( F2 @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) ) @ H2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% execute_map_entry(1)
thf(fact_4863_upd__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd @ A )
= ( ^ [I2: nat,X2: A,A5: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H @ A5 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I2 @ X2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% upd_def
thf(fact_4864_execute__upd_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A4 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A4 @ I @ X @ H2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_upd(1)
thf(fact_4865_execute__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F2: nat > A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F2 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) ) ) ) ).
% execute_make
thf(fact_4866_execute__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
@ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ) ).
% execute_of_list
thf(fact_4867_execute__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) ) ) ) ).
% execute_new
thf(fact_4868_new__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N2: nat,X2: A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N2 @ X2 ) @ H ) ) ) ) ) ) ).
% new_def
thf(fact_4869_of__list__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
@ ( array_alloc @ A @ Xs3 @ H ) ) ) ) ) ) ).
% of_list_def
thf(fact_4870_Array__Time_Omake__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_make @ A )
= ( ^ [N2: nat,F: nat > A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) @ H ) ) ) ) ) ) ).
% Array_Time.make_def
thf(fact_4871_effect__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H5: heap_ext @ product_unit,F2: nat > A,N: nat,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A4 @ H5 )
= ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F2 ) @ H2 @ H5 @ A4 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_makeI
thf(fact_4872_effect__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H5: heap_ext @ product_unit,N: nat,X: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A4 @ H5 )
= ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H5 @ A4 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_newI
thf(fact_4873_effect__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H5: heap_ext @ product_unit,Xs: list @ A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A4 @ H5 )
= ( array_alloc @ A @ Xs @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H5 @ A4 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% effect_of_listI
thf(fact_4874_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C5: heap_Time_Heap @ A,H: heap_ext @ product_unit,H6: heap_ext @ product_unit,R2: A,N2: nat] :
( ( heap_Time_execute @ A @ C5 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N2 ) ) ) ) ) ) ).
% effect_def
thf(fact_4875_effectI,axiom,
! [A: $tType,C3: heap_Time_Heap @ A,H2: heap_ext @ product_unit,R3: A,H5: heap_ext @ product_unit,N: nat] :
( ( ( heap_Time_execute @ A @ C3 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N ) ) ) )
=> ( heap_Time_effect @ A @ C3 @ H2 @ H5 @ R3 @ N ) ) ).
% effectI
thf(fact_4876_Array__Time_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_alloc @ A )
= ( ^ [Xs3: list @ A,H: heap_ext @ product_unit] :
( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H ) )
@ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H ) ) @ Xs3
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
@ H ) ) ) ) ) ) ).
% Array_Time.alloc_def
thf(fact_4877_Heap__lub__empty,axiom,
! [A: $tType] :
( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
= ( heap_Time_Heap2 @ A
@ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_lub_empty
thf(fact_4878_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A4: A,B3: B,R: set @ ( product_prod @ A @ B ),F2: A > C,G2: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ A4 ) @ ( G2 @ B3 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F2 @ G2 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_4879_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G2: D > B,A4: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 @ ( product_Pair @ C @ D @ A4 @ B3 ) )
= ( product_Pair @ A @ B @ ( F2 @ A4 ) @ ( G2 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_4880_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F2: C > A,G2: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F2 @ G2 ) @ R ) )
=> ~ ! [X3: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F2 @ X3 ) @ ( G2 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_4881_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 )
@ ^ [X2: A,Y2: B] : ( product_Pair @ C @ D @ ( F @ X2 ) @ ( G @ Y2 ) ) ) ) ) ).
% map_prod_def
thf(fact_4882_Ref__Time_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_alloc @ A )
= ( ^ [X2: A,H: heap_ext @ product_unit] :
( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( ref2 @ A @ ( lim @ product_unit @ H ) )
@ ( ref_set @ A @ ( ref2 @ A @ ( lim @ product_unit @ H ) ) @ X2
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
@ H ) ) ) ) ) ) ).
% Ref_Time.alloc_def
thf(fact_4883_bot__in__iterates,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] : ( member @ A @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( comple6359979572994053840erates @ A @ F2 ) ) ) ).
% bot_in_iterates
thf(fact_4884_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F: ( product_prod @ B @ C ) > A,A5: B,B4: C] : ( F @ ( product_Pair @ B @ C @ A5 @ B4 ) ) ) ) ).
% curry_conv
thf(fact_4885_curryI,axiom,
! [A: $tType,B: $tType,F2: ( product_prod @ A @ B ) > $o,A4: A,B3: B] :
( ( F2 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F2 @ A4 @ B3 ) ) ).
% curryI
thf(fact_4886_curryE,axiom,
! [A: $tType,B: $tType,F2: ( product_prod @ A @ B ) > $o,A4: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F2 @ A4 @ B3 )
=> ( F2 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ).
% curryE
thf(fact_4887_curryD,axiom,
! [A: $tType,B: $tType,F2: ( product_prod @ A @ B ) > $o,A4: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F2 @ A4 @ B3 )
=> ( F2 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ).
% curryD
thf(fact_4888_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C5: ( product_prod @ A @ B ) > C,X2: A,Y2: B] : ( C5 @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) ) ) ).
% curry_def
thf(fact_4889_effect__refI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,H5: heap_ext @ product_unit,V: A,H2: heap_ext @ product_unit,N: nat] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R3 @ H5 )
= ( ref_alloc @ A @ V @ H2 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H2 @ H5 @ R3 @ N ) ) ) ) ).
% effect_refI
thf(fact_4890_ref__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_ref @ A )
= ( ^ [V2: A] :
( heap_Time_heap @ ( ref @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: ref @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( one_one @ nat ) ) )
@ ( ref_alloc @ A @ V2 @ H ) ) ) ) ) ) ).
% ref_def
thf(fact_4891_execute__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H2 )
= ( some @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: ref @ A,H6: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( one_one @ nat ) ) )
@ ( ref_alloc @ A @ V @ H2 ) ) ) ) ) ).
% execute_ref
thf(fact_4892_next__fresh,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,H5: heap_ext @ product_unit,X: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R3 @ H5 )
= ( ref_alloc @ A @ X @ H2 ) )
=> ~ ( ref_present @ A @ H2 @ R3 ) ) ) ).
% next_fresh
thf(fact_4893_next__present,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,H5: heap_ext @ product_unit,X: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R3 @ H5 )
= ( ref_alloc @ A @ X @ H2 ) )
=> ( ref_present @ A @ H5 @ R3 ) ) ) ).
% next_present
thf(fact_4894_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 ) ) )
= ( ? [N5: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ N5 @ M )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( P @ ( product_Pair @ nat @ nat @ N2 @ M ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_4895_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_4896_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_4897_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_4898_Nitpick_OEx1__unfold,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P3: A > $o] :
? [X2: A] :
( ( collect @ A @ P3 )
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Nitpick.Ex1_unfold
thf(fact_4899_adm__wf__def,axiom,
! [B: $tType,A: $tType] :
( ( adm_wf @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
! [F: A > B,G: A > B,X2: A] :
( ! [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X2 ) @ R6 )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( F7 @ F @ X2 )
= ( F7 @ G @ X2 ) ) ) ) ) ).
% adm_wf_def
thf(fact_4900_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ~ ( P @ X2 )
| ( Q @ X2 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q @ F5 ) ) ) ) ).
% eventually_frequentlyE
thf(fact_4901_frequently__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( frequently @ A
@ ^ [X2: A] : P
@ F5 )
= P ) ) ).
% frequently_const
thf(fact_4902_frequently__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] : P
@ F5 )
= ( P
& ( F5
!= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% frequently_const_iff
thf(fact_4903_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_4904_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.insert_remove
thf(fact_4905_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A3 )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.remove
thf(fact_4906_transp__trans__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X2: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 ) )
= ( trans @ A @ R3 ) ) ).
% transp_trans_eq
thf(fact_4907_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_4908_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( F2 @ X3 @ Y3 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_neutr_set.closed
thf(fact_4909_semilattice__neutr__set__def,axiom,
! [A: $tType] :
( ( lattic5652469242046573047tr_set @ A )
= ( semilattice_neutr @ A ) ) ).
% semilattice_neutr_set_def
thf(fact_4910_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_4911_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_4912_Fpow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_Fpow @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_4913_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_4914_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 ) )
=> ~ ! [I3: code_natural,J2: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( code_Suc @ I3 ) @ J2 ) ) ) ).
% random_aux_set.cases
thf(fact_4915_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ C,G2: ( sum_sum @ B @ C ) > A] :
( ( finite_finite @ B @ A3 )
=> ( ( finite_finite @ C @ B2 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G2 @ ( sum_Plus @ B @ C @ A3 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G2 @ ( sum_Inl @ B @ C ) ) @ A3 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G2 @ ( sum_Inr @ C @ B ) ) @ B2 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_4916_empty__in__Fpow,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ).
% empty_in_Fpow
thf(fact_4917_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_4918_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_4919_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_4920_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_4921_Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Node @ B @ A )
= ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
@ ^ [P7: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
? [F: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat,K4: nat] :
( ( P7
= ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F @ X2 ) )
& ( ( F @ K4 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% Node_def
thf(fact_4922_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 ) )
@ ^ [Z3: D] : ( insert2 @ D @ Z3 @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% sum_set_defs(2)
thf(fact_4923_sum__set__defs_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( basic_setl @ A @ B )
= ( sum_case_sum @ A @ ( set @ A ) @ B
@ ^ [Z3: A] : ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [B4: B] : ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_defs(1)
thf(fact_4924_Node__K0__I,axiom,
! [B: $tType,A: $tType,A4: 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 ) )
@ A4 )
@ ( old_Node @ A @ B ) ) ).
% Node_K0_I
thf(fact_4925_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_4926_Atom__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Atom @ A @ B )
= ( ^ [X2: 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 ) )
@ X2 ) )
@ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% Atom_def
thf(fact_4927_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 ) ) )
=> ( ! [N4: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N4 ) )
=> ( ! [A10: nat,Aa4: nat] :
( ( ( nat_sum_decode @ N4 )
= ( sum_Inl @ nat @ nat @ A10 ) )
=> ( ( ( nat_sum_decode @ A10 )
= ( sum_Inl @ nat @ nat @ Aa4 ) )
=> ( P @ Aa4 ) ) )
=> ( ! [A10: nat,B10: nat] :
( ( ( nat_sum_decode @ N4 )
= ( sum_Inl @ nat @ nat @ A10 ) )
=> ( ( ( nat_sum_decode @ A10 )
= ( sum_Inr @ nat @ nat @ B10 ) )
=> ( P @ B10 ) ) )
=> ( ! [B10: nat,Ba2: nat,X4: nat,Y5: nat] :
( ( ( nat_sum_decode @ N4 )
= ( sum_Inr @ nat @ nat @ B10 ) )
=> ( ( ( nat_sum_decode @ B10 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X4 @ Y5 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P @ X4 ) ) ) )
=> ( ! [B10: nat,Ba2: nat,X4: nat,Y5: nat] :
( ( ( nat_sum_decode @ N4 )
= ( sum_Inr @ nat @ nat @ B10 ) )
=> ( ( ( nat_sum_decode @ B10 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X4 @ Y5 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P @ Y5 ) ) ) )
=> ( P @ ( suc @ N4 ) ) ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% nth_item.pinduct
thf(fact_4928_ntrunc__0,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( zero_zero @ nat ) @ M4 )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_0
thf(fact_4929_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_4930_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_4931_dsum__In1I,axiom,
! [B: $tType,A: $tType,N3: set @ ( old_node @ A @ B ),N9: set @ ( old_node @ A @ B ),S2: 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 ) ) @ N3 @ N9 ) @ S2 )
=> ( 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 @ N3 ) @ ( old_In1 @ A @ B @ N9 ) ) @ ( old_dsum @ A @ B @ R3 @ S2 ) ) ) ).
% dsum_In1I
thf(fact_4932_is__empty__bot,axiom,
! [A: $tType] : ( is_empty @ A @ ( bot_bot @ ( pred @ A ) ) ) ).
% is_empty_bot
thf(fact_4933_dsum__In0I,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B ),M8: set @ ( old_node @ A @ B ),R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M4 @ M8 ) @ 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 @ M8 ) ) @ ( old_dsum @ A @ B @ R3 @ S2 ) ) ) ).
% dsum_In0I
thf(fact_4934_dsum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dsum @ A @ B )
= ( ^ [R2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S5: 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 ) ) ) )
@ ^ [X2: 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 @ X2 ) @ ( old_In0 @ A @ B @ X9 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ R2 ) )
@ ( 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 ) ) ) )
@ ^ [Y2: 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 @ Y2 ) @ ( old_In1 @ A @ B @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S5 ) ) ) ) ) ).
% dsum_def
thf(fact_4935_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 ) ) ),S2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ W2 @ ( old_dsum @ A @ B @ R3 @ S2 ) )
=> ( ! [X3: set @ ( old_node @ A @ B ),X10: 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 ) ) @ X3 @ X10 ) @ R3 )
=> ( W2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X3 ) @ ( old_In0 @ A @ B @ X10 ) ) ) )
=> ~ ! [Y3: set @ ( old_node @ A @ B ),Y7: 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 ) ) @ Y3 @ Y7 ) @ S2 )
=> ( W2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y3 ) @ ( old_In1 @ A @ B @ Y7 ) ) ) ) ) ) ).
% dsumE
thf(fact_4936_Predicate_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: pred @ A] :
( A8
= ( bot_bot @ ( pred @ A ) ) ) ) ) ).
% Predicate.is_empty_def
thf(fact_4937_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_4938_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P3: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_4939_subset_Osuc__Union__closed__empty,axiom,
! [A: $tType,A3: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) ) ) ).
% subset.suc_Union_closed_empty
thf(fact_4940_pred__on_Osuc__Union__closed__empty,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ A @ A3 @ P ) ) ).
% pred_on.suc_Union_closed_empty
thf(fact_4941_dprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dprod @ A @ B )
= ( ^ [R2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S5: 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 ) ) ) )
@ ^ [X2: 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 ) ) ) )
@ ^ [Y2: 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 @ X2 @ Y2 ) @ ( old_Scons @ A @ B @ X9 @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S5 ) ) )
@ R2 ) ) ) ) ).
% dprod_def
thf(fact_4942_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_4943_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 ),L12: list @ A,L23: 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 ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls )
=> ( P @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_4944_dprodI,axiom,
! [B: $tType,A: $tType,M4: set @ ( old_node @ A @ B ),M8: set @ ( old_node @ A @ B ),R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),N3: set @ ( old_node @ A @ B ),N9: set @ ( old_node @ A @ B ),S2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M4 @ M8 ) @ 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 ) ) @ N3 @ N9 ) @ S2 )
=> ( 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 @ N3 ) @ ( old_Scons @ A @ B @ M8 @ N9 ) ) @ ( old_dprod @ A @ B @ R3 @ S2 ) ) ) ) ).
% dprodI
thf(fact_4945_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_4946_dprodE,axiom,
! [B: $tType,A: $tType,C3: 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 ) ) ),S2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ C3 @ ( old_dprod @ A @ B @ R3 @ S2 ) )
=> ~ ! [X3: set @ ( old_node @ A @ B ),Y3: set @ ( old_node @ A @ B ),X10: 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 ) ) @ X3 @ X10 ) @ R3 )
=> ! [Y7: 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 ) ) @ Y3 @ Y7 ) @ S2 )
=> ( C3
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X3 @ Y3 ) @ ( old_Scons @ A @ B @ X10 @ Y7 ) ) ) ) ) ) ).
% dprodE
thf(fact_4947_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 ),L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_4948_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_4949_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_4950_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( order_strict_mono @ A @ B @ F2 ) ) ) ).
% strict_monoI
thf(fact_4951_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F: A > B] :
! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_4952_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_4953_merge__list_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( 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 ) ) ) ) ) ) ) )
=> ~ ! [L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ 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 ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_4954_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc23: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( 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 ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) )
=> ( ( merge_list @ A @ Acc23 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc23 ) @ Ls2 ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_4955_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_4956_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_4957_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_4958_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_4959_uprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_uprod @ A @ B )
= ( ^ [A8: set @ ( set @ ( old_node @ A @ B ) ),B7: 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 ) ) )
@ ^ [X2: 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 ) ) )
@ ^ [Y2: set @ ( old_node @ A @ B )] : ( insert2 @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y2 ) @ ( bot_bot @ ( set @ ( set @ ( old_node @ A @ B ) ) ) ) )
@ B7 ) )
@ A8 ) ) ) ) ).
% uprod_def
thf(fact_4960_heap__step__admissible,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: E > A > C > B > D > $o,X: E] :
( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) )
@ ^ [F: A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) )] :
! [H: A,H6: C,R2: B,N2: D] :
( ( ( F @ H )
= ( some @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ R2 @ ( product_Pair @ C @ D @ H6 @ N2 ) ) ) )
=> ( P @ X @ H @ H6 @ R2 @ N2 ) ) ) ).
% heap_step_admissible
thf(fact_4961_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_4962_partial__function__definitions_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,Leq: A > A > $o,Lub: ( set @ A ) > A,U3: C > B > A,F5: C > C,C2: ( B > A ) > C,F2: C,P: ( B > A ) > $o] :
( ( partia7178651479351089652itions @ A @ Leq @ Lub )
=> ( ! [X3: B] :
( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ Leq
@ ^ [F: B > A] : ( U3 @ ( F5 @ ( C2 @ F ) ) @ X3 ) )
=> ( ( F2
= ( C2
@ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq )
@ ^ [F: B > A] : ( U3 @ ( F5 @ ( C2 @ F ) ) ) ) ) )
=> ( ! [F4: B > A] :
( ( U3 @ ( C2 @ F4 ) )
= F4 )
=> ( ( comple1908693960933563346ssible @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ P )
=> ( ( P
@ ^ [Uu: B] : ( Lub @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [F4: C] :
( ( P @ ( U3 @ F4 ) )
=> ( P @ ( U3 @ ( F5 @ F4 ) ) ) )
=> ( P @ ( U3 @ F2 ) ) ) ) ) ) ) ) ) ).
% partial_function_definitions.fixp_induct_uc
thf(fact_4963_in__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_in_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_4964_ref_Oset,axiom,
! [A: $tType,X: nat] :
( ( set_ref @ A @ ( ref2 @ A @ X ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ref.set
thf(fact_4965_array_Oset,axiom,
! [A: $tType,X: nat] :
( ( set_array @ A @ ( array2 @ A @ X ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% array.set
thf(fact_4966_wfrecI,axiom,
! [B: $tType,A: $tType,X: A,R: set @ ( product_prod @ A @ A ),F5: ( A > B ) > A > B,G2: A > B] :
( ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X ) @ R )
=> ( wfrec_rel @ A @ B @ R @ F5 @ Z4 @ ( G2 @ Z4 ) ) )
=> ( wfrec_rel @ A @ B @ R @ F5 @ X @ ( F5 @ G2 @ X ) ) ) ).
% wfrecI
thf(fact_4967_wfrec__rel_Ocases,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F5: ( A > B ) > A > B,A1: A,A22: B] :
( ( wfrec_rel @ A @ B @ R @ F5 @ A1 @ A22 )
=> ~ ! [G4: A > B] :
( ( A22
= ( F5 @ G4 @ A1 ) )
=> ~ ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ A1 ) @ R )
=> ( wfrec_rel @ A @ B @ R @ F5 @ Z6 @ ( G4 @ Z6 ) ) ) ) ) ).
% wfrec_rel.cases
thf(fact_4968_wfrec__rel_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( wfrec_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B,A12: A,A23: B] :
? [X2: A,G: A > B] :
( ( A12 = X2 )
& ( A23
= ( F7 @ G @ X2 ) )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X2 ) @ R6 )
=> ( wfrec_rel @ A @ B @ R6 @ F7 @ Y2 @ ( G @ Y2 ) ) ) ) ) ) ).
% wfrec_rel.simps
thf(fact_4969_comp__fun__idem__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( finite_comp_fun_idem @ A @ A @ ( sup_sup @ A ) ) ) ).
% comp_fun_idem_sup
% Type constructors (802)
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( heap_Time_Heap @ A19 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( heap_Time_Heap @ A19 ) ) ) ).
thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
code_term_of @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
typerep2 @ code_natural ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
typerep2 @ code_integer ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
code_term_of @ code_term ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
typerep2 @ code_term ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( heap_ext @ A19 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( heap_ext @ A19 ) ) ) ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
typerep2 @ product_unit ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( code_term_of @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( enum @ A20 ) )
=> ( enum @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( typerep2 @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_14,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( typerep2 @ ( old_node @ A19 @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_16,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( multiset @ A19 ) ) ) ).
thf(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17,axiom,
code_term_of @ typerep ).
thf(tcon_Typerep_Otyperep___Typerep_Otyperep_18,axiom,
typerep2 @ typerep ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
typerep2 @ assn ).
thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_20,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( pred @ A19 ) ) ) ).
thf(tcon_Predicate_Opred___Typerep_Otyperep_21,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( pred @ A19 ) ) ) ).
thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_22,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( seq @ A19 ) ) ) ).
thf(tcon_Predicate_Oseq___Typerep_Otyperep_23,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( seq @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_24,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_25,axiom,
! [A19: $tType] :
( ( enum @ A19 )
=> ( enum @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_26,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( option @ A19 ) ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_27,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( filter @ A19 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_28,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( filter @ A19 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_29,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( code_term_of @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_30,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( enum @ A20 ) )
=> ( enum @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_31,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( typerep2 @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_32,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( array @ A19 ) ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_33,axiom,
! [A19: $tType] : ( typerep2 @ ( array @ A19 ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_34,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_35,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( list @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_36,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( ref @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_37,axiom,
! [A19: $tType] : ( typerep2 @ ( ref @ A19 ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_38,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_39,axiom,
enum @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_40,axiom,
typerep2 @ $o ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_41,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_42,axiom,
! [A19: $tType] :
( ( enum @ A19 )
=> ( enum @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_43,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( set @ A19 ) ) ) ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_44,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_45,axiom,
typerep2 @ rat ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_46,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_47,axiom,
typerep2 @ num ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_48,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_49,axiom,
typerep2 @ nat ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_50,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Typerep_Otyperep_51,axiom,
typerep2 @ int ).
thf(tcon_itself___Code__Evaluation_Oterm__of_52,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( code_term_of @ ( itself @ A19 ) ) ) ).
thf(tcon_itself___Typerep_Otyperep_53,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( typerep2 @ ( itself @ A19 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_54,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( code_term_of @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Enum_Oenum_55,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( enum @ A20 ) )
=> ( enum @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep_56,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep2 @ A19 )
& ( typerep2 @ A20 ) )
=> ( typerep2 @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A19: $tType,A20: $tType] :
( ( ( cl_HOL_Oequal @ A19 )
& ( quickc3360725361186068524ustive @ A19 )
& ( quickc3360725361186068524ustive @ A20 ) )
=> ( quickc3360725361186068524ustive @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4346867609351753570nf_top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A19: $tType,A20: $tType] :
( ( ( cl_HOL_Oequal @ A19 )
& ( quickc658316121487927005ustive @ A19 )
& ( quickc658316121487927005ustive @ A20 ) )
=> ( quickc658316121487927005ustive @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A19: $tType,A20: $tType] :
( ( boolea8198339166811842893lgebra @ A20 )
=> ( boolea8198339166811842893lgebra @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A19: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A19: $tType,A20: $tType] :
( ( ( code_term_of @ A19 )
& ( cl_HOL_Oequal @ A19 )
& ( quickcheck_random @ A20 ) )
=> ( quickcheck_random @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A19: $tType,A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A19: $tType,A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A19: $tType,A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A19: $tType,A20: $tType] :
( ( order_bot @ A20 )
=> ( order_bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A19: $tType,A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A19: $tType,A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A19: $tType,A20: $tType] :
( ( order @ A20 )
=> ( order @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A19: $tType,A20: $tType] :
( ( top @ A20 )
=> ( top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A19: $tType,A20: $tType] :
( ( ord @ A20 )
=> ( ord @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A19: $tType,A20: $tType] :
( ( bot @ A20 )
=> ( bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A19: $tType,A20: $tType] :
( ( uminus @ A20 )
=> ( uminus @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A19: $tType,A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( sup @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A19: $tType,A20: $tType] :
( ( minus @ A20 )
=> ( minus @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( cl_HOL_Oequal @ A20 ) )
=> ( cl_HOL_Oequal @ ( A19 > A20 ) ) ) ).
thf(tcon_itself___Quickcheck__Random_Orandom_57,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( quickcheck_random @ ( itself @ A19 ) ) ) ).
thf(tcon_itself___HOL_Oequal_58,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( itself @ A19 ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_59,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_60,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_61,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_62,axiom,
quickcheck_random @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_63,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_64,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_65,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___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___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___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_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___Lattices_Osup_71,axiom,
sup @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_72,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___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Int_Oint___HOL_Oequal_73,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_74,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_75,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_76,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_77,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_78,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_79,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_80,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_81,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_82,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_83,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_84,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_85,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_86,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_87,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_88,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_89,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_90,axiom,
quickc3360725361186068524ustive @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_91,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_92,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_93,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_94,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_95,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_96,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_97,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_98,axiom,
quickc658316121487927005ustive @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_99,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_100,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_101,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_102,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_103,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_104,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_105,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_106,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_107,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_108,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_109,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_110,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_111,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_112,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_113,axiom,
quickcheck_random @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_114,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_115,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_116,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_117,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_118,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_119,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_120,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_121,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_122,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_123,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_124,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_125,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_126,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_127,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_128,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_129,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_130,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_131,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_132,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_133,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_134,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_135,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_136,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_137,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_138,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_139,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_140,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_141,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_142,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_143,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_144,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_145,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_146,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_147,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_148,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_149,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_150,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_151,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Lattices_Osup_152,axiom,
sup @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_153,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_154,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_155,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_156,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_157,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_158,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_159,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_160,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_161,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_162,axiom,
quickc3360725361186068524ustive @ num ).
thf(tcon_Num_Onum___Quickcheck__Random_Orandom_163,axiom,
quickcheck_random @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_164,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_165,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_166,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_167,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_168,axiom,
times @ num ).
thf(tcon_Num_Onum___HOL_Oequal_169,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_170,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_171,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_172,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_173,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_174,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_175,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_176,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_177,axiom,
quickc3360725361186068524ustive @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_178,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_179,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_180,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_181,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_182,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_183,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_184,axiom,
quickc658316121487927005ustive @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_185,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_186,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_187,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_188,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_189,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_190,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_191,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_192,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_193,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_194,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_195,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_196,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_197,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_198,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_199,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_200,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_201,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_202,axiom,
quickcheck_random @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_203,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_204,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_205,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_206,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_207,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_208,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_209,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_210,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_211,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_212,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_213,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_214,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_215,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_216,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_217,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_218,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_219,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_220,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_221,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_222,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_223,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_224,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_225,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_226,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_227,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_228,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_229,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_230,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_231,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_232,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_233,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_234,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_235,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_236,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_237,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_238,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_239,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_240,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_241,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_242,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_243,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_244,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_245,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_246,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Lattices_Osup_247,axiom,
sup @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_248,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_249,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_250,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_251,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_252,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_253,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_254,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_255,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_256,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_257,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_258,axiom,
! [A19: $tType] : ( condit1219197933456340205attice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_259,axiom,
! [A19: $tType] : ( comple592849572758109894attice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_260,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_261,axiom,
! [A19: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_262,axiom,
! [A19: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_263,axiom,
! [A19: $tType] : ( comple6319245703460814977attice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_264,axiom,
! [A19: $tType] :
( ( quickc658316121487927005ustive @ A19 )
=> ( quickc658316121487927005ustive @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_265,axiom,
! [A19: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_266,axiom,
! [A19: $tType] : ( bounded_lattice_top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_267,axiom,
! [A19: $tType] : ( bounded_lattice_bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_268,axiom,
! [A19: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_269,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_270,axiom,
! [A19: $tType] : ( semilattice_sup @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_271,axiom,
! [A19: $tType] : ( semilattice_inf @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_272,axiom,
! [A19: $tType] : ( distrib_lattice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_273,axiom,
! [A19: $tType] : ( bounded_lattice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_274,axiom,
! [A19: $tType] : ( order_top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_275,axiom,
! [A19: $tType] : ( order_bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_276,axiom,
! [A19: $tType] : ( preorder @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_277,axiom,
! [A19: $tType] : ( lattice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_278,axiom,
! [A19: $tType] : ( order @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_279,axiom,
! [A19: $tType] : ( top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_280,axiom,
! [A19: $tType] : ( ord @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_281,axiom,
! [A19: $tType] : ( bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_282,axiom,
! [A19: $tType] : ( uminus @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_283,axiom,
! [A19: $tType] : ( sup @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_284,axiom,
! [A19: $tType] : ( minus @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_285,axiom,
! [A19: $tType] :
( ( cl_HOL_Oequal @ A19 )
=> ( cl_HOL_Oequal @ ( set @ A19 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_286,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_287,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_288,axiom,
quickc3360725361186068524ustive @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_289,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_290,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_291,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_292,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_293,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_294,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_295,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_296,axiom,
quickcheck_random @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_297,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_298,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_299,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_300,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_301,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_302,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_303,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_304,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_305,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_306,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_307,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_308,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_309,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_310,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_311,axiom,
sup @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_312,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_313,axiom,
heap @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_314,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_315,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( quickc3360725361186068524ustive @ ( ref @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_316,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( quickcheck_random @ ( ref @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_317,axiom,
! [A19: $tType] : ( heap @ ( ref @ A19 ) ) ).
thf(tcon_Heap_Oref___HOL_Oequal_318,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( ref @ A19 ) ) ).
thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_319,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___Quickcheck__Random_Orandom_320,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_321,axiom,
! [A19: $tType] :
( ( heap @ A19 )
=> ( heap @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_322,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( list @ A19 ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_323,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( quickc3360725361186068524ustive @ ( array @ A19 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_324,axiom,
! [A19: $tType] :
( ( typerep2 @ A19 )
=> ( quickcheck_random @ ( array @ A19 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_325,axiom,
! [A19: $tType] : ( heap @ ( array @ A19 ) ) ).
thf(tcon_Heap_Oarray___HOL_Oequal_326,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( array @ A19 ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_327,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickc3360725361186068524ustive @ A19 )
& ( quickc3360725361186068524ustive @ A20 ) )
=> ( quickc3360725361186068524ustive @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_328,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickcheck_random @ A19 )
& ( quickcheck_random @ A20 ) )
=> ( quickcheck_random @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_329,axiom,
! [A19: $tType,A20: $tType] :
( ( ( heap @ A19 )
& ( heap @ A20 ) )
=> ( heap @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_330,axiom,
! [A19: $tType,A20: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A19 @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_331,axiom,
! [A19: $tType] : ( condit1219197933456340205attice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_332,axiom,
! [A19: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_333,axiom,
! [A19: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_334,axiom,
! [A19: $tType] : ( comple6319245703460814977attice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_335,axiom,
! [A19: $tType] : ( bounded_lattice_top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_336,axiom,
! [A19: $tType] : ( bounded_lattice_bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_337,axiom,
! [A19: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_338,axiom,
! [A19: $tType] : ( semilattice_sup @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_339,axiom,
! [A19: $tType] : ( semilattice_inf @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_340,axiom,
! [A19: $tType] : ( distrib_lattice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_341,axiom,
! [A19: $tType] : ( bounded_lattice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_342,axiom,
! [A19: $tType] : ( order_top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_343,axiom,
! [A19: $tType] : ( order_bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_344,axiom,
! [A19: $tType] : ( preorder @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_345,axiom,
! [A19: $tType] : ( lattice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_346,axiom,
! [A19: $tType] : ( order @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_347,axiom,
! [A19: $tType] : ( top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_348,axiom,
! [A19: $tType] : ( ord @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_349,axiom,
! [A19: $tType] : ( bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osup_350,axiom,
! [A19: $tType] : ( sup @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_351,axiom,
! [A19: $tType] :
( ( cl_HOL_Oequal @ A19 )
=> ( cl_HOL_Oequal @ ( filter @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_352,axiom,
! [A19: $tType] :
( ( comple5582772986160207858norder @ A19 )
=> ( condit6923001295902523014norder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_353,axiom,
! [A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( condit1219197933456340205attice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_354,axiom,
! [A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( comple592849572758109894attice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_355,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_356,axiom,
! [A19: $tType] :
( ( lattice @ A19 )
=> ( bounde4967611905675639751up_bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_357,axiom,
! [A19: $tType] :
( ( bounded_lattice_top @ A19 )
=> ( bounde4346867609351753570nf_top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A19: $tType] :
( ( comple5582772986160207858norder @ A19 )
=> ( comple5582772986160207858norder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_358,axiom,
! [A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple6319245703460814977attice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_359,axiom,
! [A19: $tType] :
( ( bounded_lattice_top @ A19 )
=> ( bounded_lattice_top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_360,axiom,
! [A19: $tType] :
( ( lattice @ A19 )
=> ( bounded_lattice_bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_361,axiom,
! [A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_362,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_363,axiom,
! [A19: $tType] :
( ( semilattice_sup @ A19 )
=> ( semilattice_sup @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_364,axiom,
! [A19: $tType] :
( ( semilattice_inf @ A19 )
=> ( semilattice_inf @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_365,axiom,
! [A19: $tType] :
( ( distrib_lattice @ A19 )
=> ( distrib_lattice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_366,axiom,
! [A19: $tType] :
( ( bounded_lattice_top @ A19 )
=> ( bounded_lattice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_367,axiom,
! [A19: $tType] :
( ( wellorder @ A19 )
=> ( wellorder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_368,axiom,
! [A19: $tType] :
( ( order_top @ A19 )
=> ( order_top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_369,axiom,
! [A19: $tType] :
( ( order @ A19 )
=> ( order_bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_370,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_371,axiom,
! [A19: $tType] :
( ( linorder @ A19 )
=> ( linorder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_372,axiom,
! [A19: $tType] :
( ( lattice @ A19 )
=> ( lattice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_373,axiom,
! [A19: $tType] :
( ( order @ A19 )
=> ( order @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_374,axiom,
! [A19: $tType] :
( ( order_top @ A19 )
=> ( top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_375,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( ord @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_376,axiom,
! [A19: $tType] :
( ( order @ A19 )
=> ( bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osup_377,axiom,
! [A19: $tType] :
( ( sup @ A19 )
=> ( sup @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_378,axiom,
! [A19: $tType] :
( ( heap @ A19 )
=> ( heap @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_379,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( option @ A19 ) ) ).
thf(tcon_Predicate_Oseq___HOL_Oequal_380,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( seq @ A19 ) ) ).
thf(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_381,axiom,
! [A19: $tType] : ( condit1219197933456340205attice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_382,axiom,
! [A19: $tType] : ( comple592849572758109894attice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_383,axiom,
! [A19: $tType] : ( bounde4967611905675639751up_bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_384,axiom,
! [A19: $tType] : ( bounde4346867609351753570nf_top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_385,axiom,
! [A19: $tType] : ( comple6319245703460814977attice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_386,axiom,
! [A19: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_387,axiom,
! [A19: $tType] : ( bounded_lattice_top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_388,axiom,
! [A19: $tType] : ( bounded_lattice_bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_389,axiom,
! [A19: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_390,axiom,
! [A19: $tType] : ( semilattice_sup @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_391,axiom,
! [A19: $tType] : ( semilattice_inf @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Odistrib__lattice_392,axiom,
! [A19: $tType] : ( distrib_lattice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_393,axiom,
! [A19: $tType] : ( bounded_lattice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__top_394,axiom,
! [A19: $tType] : ( order_top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__bot_395,axiom,
! [A19: $tType] : ( order_bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Opreorder_396,axiom,
! [A19: $tType] : ( preorder @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Olattice_397,axiom,
! [A19: $tType] : ( lattice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder_398,axiom,
! [A19: $tType] : ( order @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Otop_399,axiom,
! [A19: $tType] : ( top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oord_400,axiom,
! [A19: $tType] : ( ord @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Obot_401,axiom,
! [A19: $tType] : ( bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ouminus_402,axiom,
! [A19: $tType] : ( uminus @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osup_403,axiom,
! [A19: $tType] : ( sup @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ominus_404,axiom,
! [A19: $tType] : ( minus @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___HOL_Oequal_405,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( pred @ A19 ) ) ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_406,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_407,axiom,
bounde4346867609351753570nf_top @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_408,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_409,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_410,axiom,
bounded_lattice_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_411,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_412,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_413,axiom,
distrib_lattice @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_414,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_415,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_416,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_417,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_418,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_419,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_420,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_421,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_422,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_423,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Otop_424,axiom,
top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_425,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_426,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_427,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osup_428,axiom,
sup @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_429,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_430,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_431,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_432,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_433,axiom,
dvd @ assn ).
thf(tcon_Typerep_Otyperep___Heap_Oheap_434,axiom,
heap @ typerep ).
thf(tcon_Typerep_Otyperep___HOL_Oequal_435,axiom,
cl_HOL_Oequal @ typerep ).
thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_436,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_437,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_438,axiom,
! [A19: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_439,axiom,
! [A19: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_440,axiom,
! [A19: $tType] : ( cancel_semigroup_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_441,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_442,axiom,
! [A19: $tType] : ( comm_monoid_diff @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_443,axiom,
! [A19: $tType] : ( ab_semigroup_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_444,axiom,
! [A19: $tType] : ( comm_monoid_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_445,axiom,
! [A19: $tType] : ( semigroup_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_446,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_447,axiom,
! [A19: $tType] : ( monoid_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_448,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( order @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_449,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( ord @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_450,axiom,
! [A19: $tType] : ( minus @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_451,axiom,
! [A19: $tType] : ( zero @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_452,axiom,
! [A19: $tType] :
( ( cl_HOL_Oequal @ A19 )
=> ( cl_HOL_Oequal @ ( multiset @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_453,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickc3360725361186068524ustive @ A19 )
& ( quickc3360725361186068524ustive @ A20 ) )
=> ( quickc3360725361186068524ustive @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_454,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickc658316121487927005ustive @ A19 )
& ( quickc658316121487927005ustive @ A20 ) )
=> ( quickc658316121487927005ustive @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_455,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickcheck_random @ A19 )
& ( quickcheck_random @ A20 ) )
=> ( quickcheck_random @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_456,axiom,
! [A19: $tType,A20: $tType] :
( ( ( heap @ A19 )
& ( heap @ A20 ) )
=> ( heap @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_457,axiom,
! [A19: $tType,A20: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A19 @ A20 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_458,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_459,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_460,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_461,axiom,
quickc3360725361186068524ustive @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_462,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_463,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_464,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_465,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_466,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_467,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_468,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_469,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_470,axiom,
quickcheck_random @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_471,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_472,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_473,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_474,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_475,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_476,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_477,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_478,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_479,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_480,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_481,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_482,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_483,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_484,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_485,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_486,axiom,
sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_487,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_488,axiom,
heap @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_489,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_490,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( heap_ext @ A19 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_491,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A19 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_492,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_493,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_494,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_495,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_496,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_497,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_498,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_499,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_500,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_501,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_502,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_503,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_504,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_505,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_506,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_507,axiom,
quickc3360725361186068524ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_508,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_509,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_510,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_511,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_512,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_513,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_514,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_515,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_516,axiom,
quickc658316121487927005ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_517,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_518,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_519,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_520,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_521,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_522,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_523,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_524,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_525,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_526,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_527,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_528,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_529,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_530,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_531,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_532,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_533,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_534,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_535,axiom,
quickcheck_random @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_536,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_537,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_538,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_539,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_540,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_541,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_542,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_543,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_544,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_545,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_546,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_547,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_548,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_549,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_550,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_551,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_552,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_553,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_554,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_555,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_556,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_557,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_558,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_559,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_560,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_561,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_562,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_563,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_564,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_565,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_566,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_567,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_568,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_569,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_570,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_571,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_572,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_573,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_574,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_575,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_576,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_577,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_578,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_579,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_580,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_581,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_582,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_583,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_584,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_585,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_586,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_587,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_588,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_589,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_590,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_591,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_592,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_593,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_594,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_595,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_596,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_597,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_598,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_599,axiom,
quickc3360725361186068524ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_600,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_601,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_602,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_603,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_604,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_605,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_606,axiom,
quickc658316121487927005ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_607,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_608,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_609,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_610,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_611,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_612,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_613,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_614,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_615,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_616,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_617,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_618,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_619,axiom,
quickcheck_random @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_620,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_621,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_622,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_623,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_624,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_625,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_626,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_627,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_628,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_629,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_630,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_631,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_632,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_633,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_634,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_635,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_636,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_637,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_638,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_639,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_640,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_641,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_642,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_643,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_644,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_645,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_646,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_647,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_648,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_649,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_650,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_651,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_652,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_653,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_654,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_655,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_656,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_657,axiom,
cl_HOL_Oequal @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_658,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( heap_Time_Heap @ A19 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_659,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A19 ) ) ).
% 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 ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
entails @ ( times_times @ assn @ ( sup_sup @ assn @ a @ b ) @ c ) @ ( sup_sup @ assn @ ( times_times @ assn @ a @ c ) @ ( times_times @ assn @ b @ c ) ) ).
%------------------------------------------------------------------------------