TPTP Problem File: ITP211^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP211^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00607_017823
% 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_00607_017823 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 7173 (2398 unt; 777 typ; 0 def)
% Number of atoms : 19288 (7368 equ; 2 cnn)
% Maximal formula atoms : 28 ( 3 avg)
% Number of connectives : 133608 (2014 ~; 216 |;1325 &;121190 @)
% ( 0 <=>;8863 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 5802 (5802 >; 0 *; 0 +; 0 <<)
% Number of symbols : 771 ( 767 usr; 7 con; 0-10 aty)
% Number of variables : 24892 (2394 ^;21114 !; 485 ?;24892 :)
% ( 899 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 14:51:34.507
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $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_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_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_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 (756)
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[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_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_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_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__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_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_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ 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_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_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_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_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_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_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : ( set @ ( product_prod @ A @ 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_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
thf(sy_c_BNF__Def_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
thf(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_BNF__Greatest__Fixpoint_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__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_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__BNF__LFPs_Octor__rec,type,
basic_BNF_ctor_rec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNF__LFPs_Oxtor,type,
basic_BNF_xtor:
!>[A: $tType] : ( 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_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_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_Basic__BNFs_Osndsp,type,
basic_sndsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B > $o ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
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_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__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_ONeg,type,
code_Neg: num > code_integer ).
thf(sy_c_Code__Numeral_OPos,type,
code_Pos: num > code_integer ).
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_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: nat > code_natural > $o ).
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__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_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Opart__equivp,type,
equiv_part_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oabstract__filter,type,
abstract_filter:
!>[A: $tType] : ( ( product_unit > ( filter @ A ) ) > ( filter @ 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_Ofilter_ORep__filter,type,
rep_filter:
!>[A: $tType] : ( ( filter @ A ) > ( A > $o ) > $o ) ).
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_Ois__filter,type,
is_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( B > C ) > 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_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_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_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
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_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_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > nat ) ).
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_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ONeg,type,
neg: num > int ).
thf(sy_c_Int_OPos,type,
pos: num > 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__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_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_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
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_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__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_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
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_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__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ 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_Onull,type,
null:
!>[A: $tType] : ( ( 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_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_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__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_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_OCODE__ABORT,type,
cODE_ABORT:
!>[A: $tType] : ( ( product_unit > A ) > A ) ).
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_Ofilter__rev,type,
filter_rev:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Ofilter__rev__aux,type,
filter_rev_aux:
!>[A: $tType] : ( ( list @ A ) > ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
thf(sy_c_Misc_Olist__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_Misc_Olist__all__zip__rel,type,
list_all_zip_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
thf(sy_c_Misc_Omap__mmupd,type,
map_mmupd:
!>[B: $tType,A: $tType] : ( ( B > ( option @ A ) ) > ( set @ B ) > A > B > ( option @ A ) ) ).
thf(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list__rel,type,
merge_list_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omerge__rel,type,
merge_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort,type,
mergesort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__split,type,
merges295452479951948502_split:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
thf(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Misc_Opartition__rev,type,
partition_rev:
!>[A: $tType] : ( ( A > $o ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Opartition__rev__rel,type,
partition_rev_rel:
!>[A: $tType] : ( ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Oquicksort__by__rel,type,
quicksort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Oquicksort__by__rel__rel,type,
quicksort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Misc_Oremove__rev,type,
remove_rev:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg__rel,type,
revg_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( option @ A ) ) ).
thf(sy_c_Misc_Oslice,type,
slice:
!>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Osu__rel__fun,type,
su_rel_fun:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_Misc_Othe__default,type,
the_default:
!>[A: $tType] : ( A > ( option @ A ) > A ) ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Misc_Ozipf,type,
zipf:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
thf(sy_c_Misc_Ozipf__rel,type,
zipf_rel:
!>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
comm_m7189776963980413722m_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mset,type,
comm_monoid_mset:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_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_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__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_Orel__mset,type,
rel_mset:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( multiset @ A ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Orepeat__mset,type,
repeat_mset:
!>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Multiset_Osize__multiset,type,
size_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
thf(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_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_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__encode,type,
nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_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_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder_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_Omono,type,
order_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_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
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_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
thf(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( A > product_unit > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
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_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_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( 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_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_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > 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_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
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_Osym,type,
sym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Osymp,type,
symp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind:
!>[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_empty:
!>[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_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_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
thf(sy_c_Sum__Type_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_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_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_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_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
thf(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_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_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_h,type,
h: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ).
% Relevant facts (5761)
thf(fact_0_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_1_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_2_Abs__assn__eqI_I2_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_3_Abs__assn__eqI_I1_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_4_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_5_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_6_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_7_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_8_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_9_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_10_equals0D,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A3 ) ) ).
% equals0D
thf(fact_11_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_12_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_13_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_14_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_15_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_16_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_17_mod__false,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H2 ) ).
% mod_false
thf(fact_18_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_19_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_20_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_21_wand__assn__def,axiom,
( wand_assn
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% wand_assn_def
thf(fact_22_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),V: B] :
( ( map_mmupd @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) @ V )
= M ) ).
% map_mmupd_empty
thf(fact_23_one__assn__def,axiom,
( ( one_one @ assn )
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_24_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_25_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_26_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( Y
= ( As2
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_27_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( As2
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_28_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( As2
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_29_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_30_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K2: set @ A,M: A > ( option @ B ),V: B] :
( ~ ( member @ A @ K @ K2 )
=> ( ( map_mmupd @ A @ B @ M @ K2 @ V @ K )
= ( M @ K ) ) ) ).
% mmupd_notin_upd
thf(fact_31_wand__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] : ( proper @ ( wand_raw @ P @ Q2 ) ) ).
% wand_proper
thf(fact_32_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_33_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_34_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F: A > B,A6: A,B2: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F @ ( product_Pair @ A @ A @ A6 @ B2 ) ) ) ).
% pairself.cases
thf(fact_35_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,B2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B2 ) @ S )
& ( P @ A6 @ B2 ) ) ) ) ).
% bex2I
thf(fact_36_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ).
% one_assn_raw.cases
thf(fact_37_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_38_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_39_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_40_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_41_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_42_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_43_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_44_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_45_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( X
= ( abs_assn @ Y2 ) )
=> ~ ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y2 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ) ) ).
% Abs_assn_cases
thf(fact_46_Abs__assn__induct,axiom,
! [P: assn > $o,X: assn] :
( ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y2 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( P @ ( abs_assn @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_assn_induct
thf(fact_47_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_48_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_ext @ product_unit,H3: 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 ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_49_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A7: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B4 ) )
= ( ( A4 = A7 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_50_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_51_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_52_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_53_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_54_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P3: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P3 ) )
= ( ? [A8: B] :
( P3
= ( product_Pair @ B @ A @ A8 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_55_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_56_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 )
=> ~ ! [H: A,As2: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
=> ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_57_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 )
=> ~ ! [H: A,As2: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
=> ~ ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_58_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 )
=> ~ ! [H: A,As2: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
=> ( Y
= ( ~ ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_59_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_60_pure__assn__proper,axiom,
! [B3: $o] : ( proper @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B3 ) ) ).
% pure_assn_proper
thf(fact_61_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,Q3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As2: 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 ) ) @ Q3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ).
% times_assn_raw.cases
thf(fact_62_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 ) ) )] :
~ ! [R: ref @ A,X3: A,H: heap_ext @ product_unit,As2: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_63_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 ) ) )] :
~ ! [R: array @ A,X3: list @ A,H: heap_ext @ product_unit,As2: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_64_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B2: $o,H: A,As2: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B2 @ ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) ) ) ).
% pure_assn_raw.cases
thf(fact_65_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B2: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_66_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y2: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_67_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A6: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_68_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A7: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B4 ) )
=> ~ ( ( A4 = A7 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_69_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B2: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_70_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B2: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_71_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,B2: B,C3: 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 ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_72_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,B2: B,C3: C,D2: D,E2: E,F: 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 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) ) ).
% prod_cases6
thf(fact_73_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
~ ! [A6: A,B2: B,C3: C,D2: D,E2: E,F: F3,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_74_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,B2: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_75_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,B2: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_76_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,B2: B,C3: 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 ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_77_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,B2: B,C3: C,D2: D,E2: E,F: 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 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_78_prod__induct7,axiom,
! [G2: $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 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
( ! [A6: A,B2: B,C3: C,D2: D,E2: E,F: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_79_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_80_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_81_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A4: B,B3: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( C2 @ A4 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_82_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F2 ) )
= F2 ) ).
% uncurry_curry_id
thf(fact_83_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C] :
( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F2 ) )
= F2 ) ).
% curry_uncurry_id
thf(fact_84_snga__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: array @ A,X: list @ A] : ( proper @ ( snga_assn_raw @ A @ R2 @ X ) ) ) ).
% snga_assn_proper
thf(fact_85_sngr__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,X: A] : ( proper @ ( sngr_assn_raw @ A @ R2 @ X ) ) ) ).
% sngr_assn_proper
thf(fact_86_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_87_times__assn__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper @ ( times_assn_raw @ P @ Q2 ) ) ) ) ).
% times_assn_proper
thf(fact_88_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn @ ( one_one @ assn ) ).
% is_pure_assn_basic_simps(2)
thf(fact_89_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y3: B,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R3 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R3 ) )
=> ( Y3 = Z3 ) )
& ! [X2: A,Y3: A,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R3 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R3 ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_90_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R4: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R4 )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_91_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_92_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_93_insertCI,axiom,
! [A: $tType,A4: A,B5: set @ A,B3: A] :
( ( ~ ( member @ A @ A4 @ B5 )
=> ( A4 = B3 ) )
=> ( member @ A @ A4 @ ( insert2 @ A @ B3 @ B5 ) ) ) ).
% insertCI
thf(fact_94_insert__iff,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ A] :
( ( member @ A @ A4 @ ( insert2 @ A @ B3 @ A3 ) )
= ( ( A4 = B3 )
| ( member @ A @ A4 @ A3 ) ) ) ).
% insert_iff
thf(fact_95_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ X @ A3 ) )
= ( insert2 @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_96_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_97_singletonI,axiom,
! [A: $tType,A4: A] : ( member @ A @ A4 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_98_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A8: B,B6: C] : ( F4 @ ( product_Pair @ B @ C @ A8 @ B6 ) ) ) ) ).
% curry_conv
thf(fact_99_insertE,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ A] :
( ( member @ A @ A4 @ ( insert2 @ A @ B3 @ A3 ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ A3 ) ) ) ).
% insertE
thf(fact_100_insertI1,axiom,
! [A: $tType,A4: A,B5: set @ A] : ( member @ A @ A4 @ ( insert2 @ A @ A4 @ B5 ) ) ).
% insertI1
thf(fact_101_insertI2,axiom,
! [A: $tType,A4: A,B5: set @ A,B3: A] :
( ( member @ A @ A4 @ B5 )
=> ( member @ A @ A4 @ ( insert2 @ A @ B3 @ B5 ) ) ) ).
% insertI2
thf(fact_102_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ~ ! [B7: set @ A] :
( ( A3
= ( insert2 @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_103_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ~ ( member @ A @ X @ B5 )
=> ( ( ( insert2 @ A @ X @ A3 )
= ( insert2 @ A @ X @ B5 ) )
= ( A3 = B5 ) ) ) ) ).
% insert_ident
thf(fact_104_insert__absorb,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( insert2 @ A @ A4 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_105_insert__eq__iff,axiom,
! [A: $tType,A4: A,A3: set @ A,B3: A,B5: set @ A] :
( ~ ( member @ A @ A4 @ A3 )
=> ( ~ ( member @ A @ B3 @ B5 )
=> ( ( ( insert2 @ A @ A4 @ A3 )
= ( insert2 @ A @ B3 @ B5 ) )
= ( ( ( A4 = B3 )
=> ( A3 = B5 ) )
& ( ( A4 != B3 )
=> ? [C4: set @ A] :
( ( A3
= ( insert2 @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B5
= ( insert2 @ A @ A4 @ C4 ) )
& ~ ( member @ A @ A4 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_106_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ Y @ A3 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_107_mk__disjoint__insert,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ? [B7: set @ A] :
( ( A3
= ( insert2 @ A @ A4 @ B7 ) )
& ~ ( member @ A @ A4 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_108_singletonD,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A4 ) ) ).
% singletonD
thf(fact_109_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_110_doubleton__eq__iff,axiom,
! [A: $tType,A4: A,B3: A,C2: A,D3: A] :
( ( ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C2 @ ( insert2 @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A4 = C2 )
& ( B3 = D3 ) )
| ( ( A4 = D3 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_111_insert__not__empty,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( insert2 @ A @ A4 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_112_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_113_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_114_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_115_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X2: A] :
( A5
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_116_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_117_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn @ ( bot_bot @ assn ) ).
% is_pure_assn_basic_simps(1)
thf(fact_118_snga__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( snga_assn @ A )
= ( ^ [R5: array @ A,A8: list @ A] : ( abs_assn @ ( snga_assn_raw @ A @ R5 @ A8 ) ) ) ) ) ).
% snga_assn_def
thf(fact_119_sngr__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( sngr_assn @ A )
= ( ^ [R5: ref @ A,X2: A] : ( abs_assn @ ( sngr_assn_raw @ A @ R5 @ X2 ) ) ) ) ) ).
% sngr_assn_def
thf(fact_120_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert2 @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_121_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_122_pure__assn__def,axiom,
( pure_assn
= ( ^ [B6: $o] : ( abs_assn @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B6 ) ) ) ) ).
% pure_assn_def
thf(fact_123_times__assn__def,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% times_assn_def
thf(fact_124_type__definition__def,axiom,
! [A: $tType,B: $tType] :
( ( type_definition @ B @ A )
= ( ^ [Rep: B > A,Abs: A > B,A5: set @ A] :
( ! [X2: B] : ( member @ A @ ( Rep @ X2 ) @ A5 )
& ! [X2: B] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( Rep @ ( Abs @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_125_type__definition_ORep__inverse,axiom,
! [A: $tType,B: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( Abs2 @ ( Rep2 @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_126_type__definition_OAbs__inverse,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,Y: A] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( Rep2 @ ( Abs2 @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_127_type__definition_ORep__inject,axiom,
! [A: $tType,B: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,X: B,Y: B] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( ( Rep2 @ X )
= ( Rep2 @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_128_type__definition_ORep__induct,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,Y: A,P: A > $o] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ! [X3: B] : ( P @ ( Rep2 @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_129_type__definition_OAbs__inject,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,X: A,Y: A] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( Abs2 @ X )
= ( Abs2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_130_pure__assn__eq__conv,axiom,
! [P: $o,Q2: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q2 ) )
= ( P = Q2 ) ) ).
% pure_assn_eq_conv
thf(fact_131_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.right_neutral
thf(fact_132_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% mult_1
thf(fact_133_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_134_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_135_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_136_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_137_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_138_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_139_is__pure__assn__pure,axiom,
! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% is_pure_assn_pure
thf(fact_140_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_141_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_142_sngr__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: ref @ A,X: A,Y: A] :
( ( times_times @ assn @ ( sngr_assn @ A @ P3 @ X ) @ ( sngr_assn @ A @ P3 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% sngr_same_false
thf(fact_143_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,X: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P3 @ X ) @ ( snga_assn @ A @ P3 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_144_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q: assn] : ( times_times @ assn @ Q @ P2 ) ) ) ).
% assn_times_comm
thf(fact_145_assn__times__assoc,axiom,
! [P: assn,Q2: assn,R4: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q2 ) @ R4 )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q2 @ R4 ) ) ) ).
% assn_times_assoc
thf(fact_146_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( times_times @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_147_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A8: A,B6: A] : ( times_times @ A @ B6 @ A8 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_148_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_149_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_150_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_151_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_152_mod__starD,axiom,
! [A3: assn,B5: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B5 ) @ 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 @ B5 @ H22 ) ) ) ).
% mod_starD
thf(fact_153_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_154_is__pure__assnE,axiom,
! [A4: assn] :
( ( is_pure_assn @ A4 )
=> ~ ! [P4: $o] :
( A4
!= ( pure_assn @ P4 ) ) ) ).
% is_pure_assnE
thf(fact_155_is__pure__assn__def,axiom,
( is_pure_assn
= ( ^ [A8: assn] :
? [P2: $o] :
( A8
= ( pure_assn @ P2 ) ) ) ) ).
% is_pure_assn_def
thf(fact_156_type__definition_ORep,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( member @ A @ ( Rep2 @ X ) @ A3 ) ) ).
% type_definition.Rep
thf(fact_157_type__definition_Ointro,axiom,
! [B: $tType,A: $tType,Rep2: B > A,A3: set @ A,Abs2: A > B] :
( ! [X3: B] : ( member @ A @ ( Rep2 @ X3 ) @ A3 )
=> ( ! [X3: B] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( Rep2 @ ( Abs2 @ Y2 ) )
= Y2 ) )
=> ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 ) ) ) ) ).
% type_definition.intro
thf(fact_158_type__definition_OAbs__cases,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ~ ! [Y2: A] :
( ( X
= ( Abs2 @ Y2 ) )
=> ~ ( member @ A @ Y2 @ A3 ) ) ) ).
% type_definition.Abs_cases
thf(fact_159_type__definition_ORep__cases,axiom,
! [A: $tType,B: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,Y: A] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ~ ! [X3: B] :
( Y
!= ( Rep2 @ X3 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_160_type__definition_OAbs__induct,axiom,
! [A: $tType,B: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,P: B > $o,X: B] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( P @ ( Abs2 @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_161_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_162_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_163_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_164_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_165_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 ) )
=> ~ ! [H: A,As2: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
=> ( ( Y
= ( ( As2
= ( 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 ) @ H @ As2 ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_166_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 ) )
=> ~ ! [H: A,As2: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
=> ( ( 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 ) @ H @ As2 ) ) )
=> ~ ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_167_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 ) )
=> ~ ! [H: A,As2: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
=> ( ( 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 ) @ H @ As2 ) ) )
=> ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_168_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A4: B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B3 ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_169_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% mult.right_assoc
thf(fact_170_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% mult.right_commute
thf(fact_171_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R5: A > A > $o,F4: B > A,X2: B,Y3: B] : ( R5 @ ( F4 @ X2 ) @ ( F4 @ Y3 ) ) ) ) ).
% in_inv_imagep
thf(fact_172_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_173_ImageI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% ImageI
thf(fact_174_Image__empty2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R4 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_175_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_176_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_177_Image__empty1,axiom,
! [B: $tType,A: $tType,X4: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_178_refl__on__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( refl_on @ A @ A3 @ ( id_on @ A @ A3 ) ) ).
% refl_on_Id_on
thf(fact_179_refl__on__domain,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ A @ A4 @ A3 )
& ( member @ A @ B3 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_180_rev__ImageI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_181_Image__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B3 ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_182_ImageE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ A3 ) )
=> ~ ! [X3: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B3 ) @ R2 )
=> ~ ( member @ B @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_183_refl__onD2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_184_refl__onD1,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_185_refl__onD,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_186_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A5: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( X2 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_187_total__onI,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( X3 != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A3 @ R2 ) ) ).
% total_onI
thf(fact_188_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_189_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_190_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_191_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_192_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( C2
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_193_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_194_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_195_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_196_proper__iff,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( relH @ As @ H2 @ H3 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ) ) ).
% proper_iff
thf(fact_197_proper__def,axiom,
( proper
= ( ^ [P2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
! [H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,As3: set @ nat] :
( ( ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) ) )
& ( ( ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) )
& ( relH @ As3 @ H4 @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As3 ) ) )
=> ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As3 ) ) ) ) ) ) ).
% proper_def
thf(fact_198_properD2,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat,H3: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( relH @ As @ H2 @ H3 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ) ) ).
% properD2
thf(fact_199_properI,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [As2: set @ nat,H: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) )
=> ( ! [As2: set @ nat,H: heap_ext @ product_unit,H6: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( relH @ As2 @ H @ H6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_200_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E3: set @ ( product_prod @ B @ A )] :
( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E3 )
= ( image @ B @ A @ E3 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_201_Range__insert,axiom,
! [A: $tType,B: $tType,A4: B,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B3 ) @ R2 ) )
= ( insert2 @ A @ B3 @ ( range2 @ B @ A @ R2 ) ) ) ).
% Range_insert
thf(fact_202_Domain__insert,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 ) )
= ( insert2 @ A @ A4 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain_insert
thf(fact_203_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_204_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_205_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_206_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_207_vimage__eq,axiom,
! [A: $tType,B: $tType,A4: A,F2: A > B,B5: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F2 @ B5 ) )
= ( member @ B @ ( F2 @ A4 ) @ B5 ) ) ).
% vimage_eq
thf(fact_208_vimageI,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: B,B3: A,B5: set @ A] :
( ( ( F2 @ A4 )
= B3 )
=> ( ( member @ A @ B3 @ B5 )
=> ( member @ B @ A4 @ ( vimage @ B @ A @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_209_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_210_Domain__Id__on,axiom,
! [A: $tType,A3: set @ A] :
( ( domain @ A @ A @ ( id_on @ A @ A3 ) )
= A3 ) ).
% Domain_Id_on
thf(fact_211_Range__Id__on,axiom,
! [A: $tType,A3: set @ A] :
( ( range2 @ A @ A @ ( id_on @ A @ A3 ) )
= A3 ) ).
% Range_Id_on
thf(fact_212_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_213_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_214_relH__sym,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ As @ H2 @ H3 )
=> ( relH @ As @ H3 @ H2 ) ) ).
% relH_sym
thf(fact_215_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_216_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > B,Q2: A > $o] :
( ! [X3: A] :
( ( P @ ( F2 @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_217_vimageI2,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: B,A3: set @ A] :
( ( member @ A @ ( F2 @ A4 ) @ A3 )
=> ( member @ B @ A4 @ ( vimage @ B @ A @ F2 @ A3 ) ) ) ).
% vimageI2
thf(fact_218_vimageE,axiom,
! [A: $tType,B: $tType,A4: A,F2: A > B,B5: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F2 @ B5 ) )
=> ( member @ B @ ( F2 @ A4 ) @ B5 ) ) ).
% vimageE
thf(fact_219_vimageD,axiom,
! [A: $tType,B: $tType,A4: A,F2: A > B,A3: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F2 @ A3 ) )
=> ( member @ B @ ( F2 @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_220_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R2 ) ) ) ) ).
% transD
thf(fact_221_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R2 ) ) ) ) ).
% transE
thf(fact_222_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_223_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R5 ) ) ) ) ) ).
% trans_def
thf(fact_224_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B2 ) @ R2 ) ) ).
% Domain.cases
thf(fact_225_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
= ( ? [A8: A,B6: B] :
( ( A4 = A8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B6 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_226_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_227_DomainE,axiom,
! [B: $tType,A: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B2 ) @ R2 ) ) ).
% DomainE
thf(fact_228_Domain__iff,axiom,
! [A: $tType,B: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_229_Range_Ocases,axiom,
! [B: $tType,A: $tType,A4: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A4 @ ( range2 @ A @ B @ R2 ) )
=> ~ ! [A6: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A4 ) @ R2 ) ) ).
% Range.cases
thf(fact_230_Range_Osimps,axiom,
! [B: $tType,A: $tType,A4: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A4 @ ( range2 @ A @ B @ R2 ) )
= ( ? [A8: A,B6: B] :
( ( A4 = B6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B6 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_231_Range_Ointros,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( range2 @ A @ B @ R2 ) ) ) ).
% Range.intros
thf(fact_232_RangeE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B3 @ ( range2 @ B @ A @ R2 ) )
=> ~ ! [A6: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B3 ) @ R2 ) ) ).
% RangeE
thf(fact_233_Range__iff,axiom,
! [A: $tType,B: $tType,A4: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A4 @ ( range2 @ B @ A @ R2 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A4 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_234_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_235_trans__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( trans @ A @ ( id_on @ A @ A3 ) ) ).
% trans_Id_on
thf(fact_236_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_237_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( ( domain @ A @ B @ R2 )
= ( bot_bot @ ( set @ A ) ) )
= ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% Domain_empty_iff
thf(fact_238_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( ( range2 @ B @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) )
= ( R2
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Range_empty_iff
thf(fact_239_relH__in__rangeI_I2_J,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ As @ H2 @ H3 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ).
% relH_in_rangeI(2)
thf(fact_240_relH__in__rangeI_I1_J,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ As @ H2 @ H3 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ).
% relH_in_rangeI(1)
thf(fact_241_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_242_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R4 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_243_mod__relH,axiom,
! [As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,P: assn] :
( ( relH @ As @ H2 @ H3 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ).
% mod_relH
thf(fact_244_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_245_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R4: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( domain @ A @ B @ R4 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R4 @ X ) ) @ R4 ) ) ).
% for_in_RI
thf(fact_246_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R2 )
=> ~ ! [B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_247_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ? [A8: A,B6: A] :
( ( A1 = A8 )
& ( A22 = B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R2 ) )
| ? [A8: A,B6: A,C5: A] :
( ( A1 = A8 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ ( transitive_trancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_248_trancl_Or__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_249_tranclE,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ~ ! [C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C3 @ B3 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_250_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_251_trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z4 ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% trancl_induct
thf(fact_252_r__r__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R4: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_253_converse__tranclE,axiom,
! [A: $tType,X: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R2 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_254_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: 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 ) @ R2 ) )
=> ( ! [A6: A,B2: 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 @ B2 ) ) @ R2 )
=> ( P @ A6 @ B2 ) )
=> ( ! [A6: A,B2: 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 @ B2 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( 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 @ B2 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A6 @ B2 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_255_converse__trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B3 ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Z4 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% converse_trancl_induct
thf(fact_256_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( P @ X3 @ Y2 ) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ X3 @ Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Y2 @ Z4 )
=> ( P @ X3 @ Z4 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_257_trancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_258_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_259_irrefl__trancl__rD,axiom,
! [A: $tType,R2: 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 @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_260_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_261_wand__assnI,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat,Q2: assn,R4: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ! [H6: heap_ext @ product_unit,As4: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As @ As4 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As @ H2 @ H6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
=> ( ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) )
=> ( rep_assn @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As4 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q2 @ R4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% wand_assnI
thf(fact_262_trancl__Image__in__Range,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),V2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ R4 ) @ V2 ) @ ( range2 @ A @ A @ R4 ) ) ).
% trancl_Image_in_Range
thf(fact_263_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_264_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),X2: B] : ( image @ B @ A @ R5 @ ( insert2 @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_265_relH__def,axiom,
( relH
= ( ^ [As3: set @ nat,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As3 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As3 ) )
& ! [T2: typerep,X2: nat] :
( ( member @ nat @ X2 @ As3 )
=> ( ( ( refs @ product_unit @ H4 @ T2 @ X2 )
= ( refs @ product_unit @ H5 @ T2 @ X2 ) )
& ( ( arrays @ product_unit @ H4 @ T2 @ X2 )
= ( arrays @ product_unit @ H5 @ T2 @ X2 ) ) ) ) ) ) ) ).
% relH_def
thf(fact_266_vimage__insert,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: B,B5: set @ B] :
( ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A4 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F2 @ B5 ) ) ) ).
% vimage_insert
thf(fact_267_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W: A,V1: A,V22: A,E3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E3 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ E3 ) )
=> ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W ) @ ( transitive_rtrancl @ A @ E3 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_268_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_269_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_270_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_271_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_272_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_273_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_274_subsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% subsetI
thf(fact_275_Int__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= ( ( member @ A @ C2 @ A3 )
& ( member @ A @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_276_IntI,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% IntI
thf(fact_277_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_278_UnCI,axiom,
! [A: $tType,C2: A,B5: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% UnCI
thf(fact_279_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_280_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_281_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_282_SigmaI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,B5: A > ( set @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ B @ B3 @ ( B5 @ A4 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_283_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) )
= ( ( member @ A @ A4 @ A3 )
& ( member @ B @ B3 @ ( B5 @ A4 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_284_Un__empty,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_285_Int__subset__iff,axiom,
! [A: $tType,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C6 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C6 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_286_Int__insert__right__if1,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( insert2 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_287_Int__insert__right__if0,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_288_insert__inter__insert,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ A3 ) @ ( insert2 @ A @ A4 @ B5 ) )
= ( insert2 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_289_Int__insert__left__if1,axiom,
! [A: $tType,A4: A,C6: set @ A,B5: set @ A] :
( ( member @ A @ A4 @ C6 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ B5 ) @ C6 )
= ( insert2 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_290_Int__insert__left__if0,axiom,
! [A: $tType,A4: A,C6: set @ A,B5: set @ A] :
( ~ ( member @ A @ A4 @ C6 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ B5 ) @ C6 )
= ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Int_insert_left_if0
thf(fact_291_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Un_subset_iff
thf(fact_292_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A4: A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( insert2 @ A @ A4 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_293_Un__insert__left,axiom,
! [A: $tType,A4: A,B5: set @ A,C6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A4 @ B5 ) @ C6 )
= ( insert2 @ A @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Un_insert_left
thf(fact_294_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_295_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_296_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( inf_inf @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_297_Un__Int__eq_I4_J,axiom,
! [A: $tType,T3: set @ A,S: set @ A] :
( ( inf_inf @ ( set @ A ) @ T3 @ ( sup_sup @ ( set @ A ) @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_298_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_299_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_300_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( sup_sup @ ( set @ A ) @ S @ ( inf_inf @ ( set @ A ) @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_301_Int__Un__eq_I4_J,axiom,
! [A: $tType,T3: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T3 @ ( inf_inf @ ( set @ A ) @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_302_vimage__Int,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B5: set @ B] :
( ( vimage @ A @ B @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B5 ) ) ) ).
% vimage_Int
thf(fact_303_vimage__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B5: set @ B] :
( ( vimage @ A @ B @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B5 ) ) ) ).
% vimage_Un
thf(fact_304_relH__dist__union,axiom,
! [As: set @ nat,As5: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ ( sup_sup @ ( set @ nat ) @ As @ As5 ) @ H2 @ H3 )
= ( ( relH @ As @ H2 @ H3 )
& ( relH @ As5 @ H2 @ H3 ) ) ) ).
% relH_dist_union
thf(fact_305_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_306_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_307_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set @ A,B3: A,B5: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ B5 ) ) )
= ( ~ ( member @ A @ B3 @ A3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_308_disjoint__insert_I1_J,axiom,
! [A: $tType,B5: set @ A,A4: A,A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B5 @ ( insert2 @ A @ A4 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A4 @ B5 )
& ( ( inf_inf @ ( set @ A ) @ B5 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_309_insert__disjoint_I2_J,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ A3 ) @ B5 ) )
= ( ~ ( member @ A @ A4 @ B5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_310_insert__disjoint_I1_J,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ A3 ) @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A4 @ B5 )
& ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_311_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_312_Image__Id__on,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A3 ) @ B5 )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ).
% Image_Id_on
thf(fact_313_in__range__dist__union,axiom,
! [H2: heap_ext @ product_unit,As: set @ nat,As5: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As @ As5 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As5 ) ) ) ) ).
% in_range_dist_union
thf(fact_314_reachable__mono,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ A @ A ),X4: set @ A,X5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ X5 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ X4 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R6 ) @ X5 ) ) ) ) ).
% reachable_mono
thf(fact_315_antisym__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 )
=> ( ( antisym @ A @ S2 )
=> ( antisym @ A @ R2 ) ) ) ).
% antisym_subset
thf(fact_316_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_317_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_318_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_319_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( A4
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_320_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_321_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_322_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S4 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_323_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_324_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ( C2 = D3 )
=> ( ord_less_eq @ A @ A4 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_325_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_326_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_327_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ A8 @ B6 )
& ( ord_less_eq @ A @ B6 @ A8 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_328_inter__eq__subsetI,axiom,
! [A: $tType,S: set @ A,S4: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ S4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ S4 )
= ( inf_inf @ ( set @ A ) @ B5 @ S4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ S )
= ( inf_inf @ ( set @ A ) @ B5 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_329_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) )
= ( inf_inf @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% Int_left_commute
thf(fact_330_Int__Collect__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B5 @ ( collect @ A @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_331_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_332_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_333_rtrancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ V2 ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_334_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% Un_left_commute
thf(fact_335_Un__Int__distrib2,axiom,
! [A: $tType,B5: set @ A,C6: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) @ A3 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C6 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_336_Un__Int__assoc__eq,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C6 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_337_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ).
% Int_left_absorb
thf(fact_338_Int__Un__distrib2,axiom,
! [A: $tType,B5: set @ A,C6: set @ A,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C6 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_339_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_left_absorb
thf(fact_340_Un__Int__distrib,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% Un_Int_distrib
thf(fact_341_Int__Un__distrib,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% Int_Un_distrib
thf(fact_342_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z5: set @ A] : Y4 = Z5 )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_343_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_344_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_345_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_346_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_347_r__le__rtrancl,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S ) ) ).
% r_le_rtrancl
thf(fact_348_subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% subset_trans
thf(fact_349_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B8 )
= B8 ) ) ) ).
% subset_Un_eq
thf(fact_350_Un__Int__crazy,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) @ ( inf_inf @ ( set @ A ) @ C6 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) @ ( sup_sup @ ( set @ A ) @ C6 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_351_Int__greatest,axiom,
! [A: $tType,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C6 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ C6 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_352_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_353_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_354_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] : ( inf_inf @ ( set @ A ) @ B8 @ A5 ) ) ) ).
% Int_commute
thf(fact_355_Int__absorb2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= A3 ) ) ).
% Int_absorb2
thf(fact_356_Int__absorb1,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_357_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_358_subset__UnE,axiom,
! [A: $tType,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ A3 )
=> ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B9 @ B5 )
=> ( C6
!= ( sup_sup @ ( set @ A ) @ A9 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_359_equalityD2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_360_equalityD1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_361_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] : ( sup_sup @ ( set @ A ) @ B8 @ A5 ) ) ) ).
% Un_commute
thf(fact_362_Un__absorb2,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= A3 ) ) ).
% Un_absorb2
thf(fact_363_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_364_Int__lower2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_365_Int__lower1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ A3 ) ).
% Int_lower1
thf(fact_366_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_367_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_368_equalityE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_369_Un__upper2,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_upper2
thf(fact_370_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_upper1
thf(fact_371_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_372_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Int_assoc
thf(fact_373_Un__least,axiom,
! [A: $tType,A3: set @ A,C6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C6 ) ) ) ).
% Un_least
thf(fact_374_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Un_assoc
thf(fact_375_Int__mono,axiom,
! [A: $tType,A3: set @ A,C6: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( inf_inf @ ( set @ A ) @ C6 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_376_subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_377_in__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_378_ball__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_379_Un__mono,axiom,
! [A: $tType,A3: set @ A,C6: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C6 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_380_bex__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B5 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_381_IntD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
=> ( member @ A @ C2 @ B5 ) ) ).
% IntD2
thf(fact_382_IntD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% IntD1
thf(fact_383_UnI2,axiom,
! [A: $tType,C2: A,B5: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% UnI2
thf(fact_384_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% UnI1
thf(fact_385_IntE,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ~ ( member @ A @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_386_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_387_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_388_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_389_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A8 )
& ( ord_less_eq @ A @ A8 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_390_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B2: A] :
( ( ord_less_eq @ A @ A6 @ B2 )
=> ( P @ A6 @ B2 ) )
=> ( ! [A6: A,B2: A] :
( ( P @ B2 @ A6 )
=> ( P @ A6 @ B2 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_391_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_392_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_393_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_394_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_395_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_396_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_397_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_398_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_399_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I: set @ A,J: set @ A,C6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I @ J ) @ C6 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C6 ) @ ( product_Sigma @ A @ B @ J @ C6 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_400_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I: set @ A,J: set @ A,C6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I @ J ) @ C6 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C6 ) @ ( product_Sigma @ A @ B @ J @ C6 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_401_Sigma__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,C6: set @ A,B5: A > ( set @ B ),D4: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X3 ) @ ( D4 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) @ ( product_Sigma @ A @ B @ C6 @ D4 ) ) ) ) ).
% Sigma_mono
thf(fact_402_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B5: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( range2 @ B @ A @ A3 ) @ ( range2 @ B @ A @ B5 ) ) ) ).
% Range_Int_subset
thf(fact_403_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B5: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( domain @ A @ B @ A3 ) @ ( domain @ A @ B @ B5 ) ) ) ).
% Domain_Int_subset
thf(fact_404_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A4: A,B3: A,P: set @ ( product_prod @ A @ A ),Q2: 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 @ Q2 ) ) )
=> ( ! [X3: A,Y2: 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 @ Y2 @ X3 ) @ Q2 )
=> ( Y2 = X3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_405_rtrancl__Un__separatorE,axiom,
! [A: $tType,A4: A,B3: A,P: set @ ( product_prod @ A @ A ),Q2: 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 @ Q2 ) ) )
=> ( ! [X3: A,Y2: 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 @ Y2 ) @ Q2 )
=> ( X3 = Y2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_406_disjoint__mono,axiom,
! [A: $tType,A4: set @ A,A7: set @ A,B3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A7 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_407_rtrancl__image__unfold__right,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),V2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ V2 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ V2 ) ) ).
% rtrancl_image_unfold_right
thf(fact_408_rtrancl__reachable__induct,axiom,
! [A: $tType,I: set @ A,INV: set @ A,E3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ I @ INV )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ INV ) @ INV )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ I ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_409_Un__Image,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( image @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R4 @ S ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R4 @ A3 ) @ ( image @ B @ A @ S @ A3 ) ) ) ).
% Un_Image
thf(fact_410_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ B @ A ),A3: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R4 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R4 @ A3 ) @ ( image @ B @ A @ R4 @ B5 ) ) ) ).
% Image_Int_subset
thf(fact_411_Image__mono,axiom,
! [B: $tType,A: $tType,R7: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A10: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R7 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R7 @ A10 ) @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% Image_mono
thf(fact_412_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B5: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A3 ) @ ( domain @ A @ B @ B5 ) ) ) ).
% Domain_Un_eq
thf(fact_413_refl__on__Un,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B5: set @ A,S2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( refl_on @ A @ B5 @ S2 )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_414_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R4 ) @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R4 ) ) ) ).
% rtrancl_sub_insert_rtrancl
thf(fact_415_Range__Un__eq,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B5: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( range2 @ B @ A @ A3 ) @ ( range2 @ B @ A @ B5 ) ) ) ).
% Range_Un_eq
thf(fact_416_refl__on__Int,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B5: set @ A,S2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( refl_on @ A @ B5 @ S2 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_417_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S2 ) ) ) ).
% Domain_mono
thf(fact_418_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 )
=> ( ord_less_eq @ ( set @ B ) @ ( range2 @ A @ B @ R2 ) @ ( range2 @ A @ B @ S2 ) ) ) ).
% Range_mono
thf(fact_419_trancl__union__outside,axiom,
! [A: $tType,V: A,W: A,E3: set @ ( product_prod @ A @ A ),U2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E3 @ U2 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E3 ) )
=> ? [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E3 @ U2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ U2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ W ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E3 @ U2 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_420_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B5 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ! [Y2: B] :
( ( member @ B @ Y2 @ ( B5 @ X3 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_421_SigmaD1,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) )
=> ( member @ A @ A4 @ A3 ) ) ).
% SigmaD1
thf(fact_422_SigmaD2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) )
=> ( member @ B @ B3 @ ( B5 @ A4 ) ) ) ).
% SigmaD2
thf(fact_423_SigmaE2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) )
=> ~ ( ( member @ A @ A4 @ A3 )
=> ~ ( member @ B @ B3 @ ( B5 @ A4 ) ) ) ) ).
% SigmaE2
thf(fact_424_trancl__image__by__rtrancl,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),Vi: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ Vi ) @ Vi )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_425_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R4 ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_426_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_427_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ? [A8: A] :
( ( A1 = A8 )
& ( A22 = A8 ) )
| ? [A8: A,B6: A,C5: A] :
( ( A1 = A8 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_428_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_429_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_430_rtranclE,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A4 != B3 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B3 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_431_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_432_rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ A4 )
=> ( ! [Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z4 ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% rtrancl_induct
thf(fact_433_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_434_converse__rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ B3 )
=> ( ! [Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ Z4 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_435_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_436_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: 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 ) @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A6: A,B2: 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 @ B2 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( 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 @ B2 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A6 @ B2 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_437_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za: A,Zb: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( 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 ) @ R2 ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A6: A,B2: 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 @ B2 ) ) @ R2 )
=> ~ ( 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 @ B2 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_438_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: 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 ) @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A6: A,B2: 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 @ B2 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( 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 ) @ R2 ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A6 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_439_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_440_Un__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_441_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B5 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_442_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_443_Int__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_444_disjoint__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ~ ( member @ A @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_445_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ~ ( member @ A @ X3 @ B5 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_446_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_447_Int__insert__right,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ( ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( insert2 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) )
& ( ~ ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_448_Int__insert__left,axiom,
! [A: $tType,A4: A,C6: set @ A,B5: set @ A] :
( ( ( member @ A @ A4 @ C6 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ B5 ) @ C6 )
= ( insert2 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) ) )
& ( ~ ( member @ A @ A4 @ C6 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A4 @ B5 ) @ C6 )
= ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) ) ) ).
% Int_insert_left
thf(fact_449_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% subrelI
thf(fact_450_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_451_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_452_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_453_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_454_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X4: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ X4 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_455_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_456_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A4: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert2 @ A @ A4 @ B5 ) ) ).
% subset_insertI
thf(fact_457_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_458_insert__mono,axiom,
! [A: $tType,C6: set @ A,D4: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A4 @ C6 ) @ ( insert2 @ A @ A4 @ D4 ) ) ) ).
% insert_mono
thf(fact_459_Image__Un,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ B @ A ),A3: set @ B,B5: set @ B] :
( ( image @ B @ A @ R4 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R4 @ A3 ) @ ( image @ B @ A @ R4 @ B5 ) ) ) ).
% Image_Un
thf(fact_460_in__rtrancl__insert,axiom,
! [A: $tType,X: product_prod @ A @ A,R4: set @ ( product_prod @ A @ A ),R2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R4 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ R2 @ R4 ) ) ) ) ).
% in_rtrancl_insert
thf(fact_461_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B,G: A > B,Y: set @ B] :
( ! [W2: A] :
( ( member @ A @ W2 @ S )
=> ( ( F2 @ W2 )
= ( G @ W2 ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_462_trancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V2 ) ) ) ) ).
% trancl_mono_mp
thf(fact_463_trancl__sub,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( transitive_trancl @ A @ R4 ) ) ).
% trancl_sub
thf(fact_464_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,B5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( vimage @ A @ B @ F2 @ B5 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ B @ ( F2 @ X2 ) @ B5 ) ) ) ) ).
% subset_vimage_iff
thf(fact_465_vimage__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F2: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ A3 ) @ ( vimage @ B @ A @ F2 @ B5 ) ) ) ).
% vimage_mono
thf(fact_466_relH__subset,axiom,
! [Bs: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,As: set @ nat] :
( ( relH @ Bs @ H2 @ H3 )
=> ( ( ord_less_eq @ ( set @ nat ) @ As @ Bs )
=> ( relH @ As @ H2 @ H3 ) ) ) ).
% relH_subset
thf(fact_467_rtrancl__apply__insert,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),X: A,S: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ ( insert2 @ A @ X @ S ) )
= ( insert2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ ( sup_sup @ ( set @ A ) @ S @ ( image @ A @ A @ R4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_468_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 )
=> ( A4 = B3 ) ) ) ) ).
% antisymD
thf(fact_469_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 )
=> ( X3 = Y2 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_470_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_471_Image__absorb__rtrancl,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ A,C6: set @ A] :
( ( trans @ A @ A3 )
=> ( ( refl_on @ A @ B5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C6 @ B5 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A3 ) @ C6 )
= ( image @ A @ A @ A3 @ C6 ) ) ) ) ) ).
% Image_absorb_rtrancl
thf(fact_472_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_473_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
( ( times_assn_raw @ P @ Q2 @ ( 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 ) )
& ( Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_474_times__assn__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( Y
= ( ~ ? [As1: set @ nat,As22: set @ nat] :
( ( As2
= ( 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 ) @ H @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_475_times__assn__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ~ ? [As12: set @ nat,As23: set @ nat] :
( ( As2
= ( 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 ) @ H @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_476_times__assn__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ? [As13: set @ nat,As24: set @ nat] :
( ( As2
= ( 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 ) @ H @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_477_antisym__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( antisym @ A @ ( id_on @ A @ A3 ) ) ).
% antisym_Id_on
thf(fact_478_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_479_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q4: A,R4: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_480_rtrancl__image__advance,axiom,
! [A: $tType,Q4: A,R4: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ R4 )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_481_trancl__rtrancl__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_482_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_483_rtrancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_484_rtrancl__into__trancl1,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_485_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R4 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_486_trancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_487_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R4 ) )
=> ? [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z4 ) @ ( transitive_rtrancl @ A @ R4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Y ) @ R4 ) ) ) ).
% tranclD2
thf(fact_488_rtranclD,axiom,
! [A: $tType,A4: A,B3: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( ( A4 = B3 )
| ( ( A4 != B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_489_tranclD,axiom,
! [A: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R4 ) )
=> ? [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z4 ) @ R4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Y ) @ ( transitive_rtrancl @ A @ R4 ) ) ) ) ).
% tranclD
thf(fact_490_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A8: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_491_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_492_singleton__Un__iff,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ( ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_493_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R4: set @ ( product_prod @ A @ A ),Y: A] :
( ~ ( member @ A @ X @ ( domain @ A @ A @ R4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R4 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_494_trancl__Image__unfold__right,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ S )
= ( image @ A @ A @ E3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ S ) ) ) ).
% trancl_Image_unfold_right
thf(fact_495_trancl__Image__unfold__left,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E3 ) @ S )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( image @ A @ A @ E3 @ S ) ) ) ).
% trancl_Image_unfold_left
thf(fact_496_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_497_subset__singleton__iff,axiom,
! [A: $tType,X4: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ X4 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ( X4
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_498_mod__star__conv,axiom,
! [A3: assn,B5: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B5 ) @ 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 @ B5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_499_star__assnI,axiom,
! [P: assn,H2: heap_ext @ product_unit,As: set @ nat,Q2: assn,As5: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As5 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As @ As5 ) ) ) ) ) ) ).
% star_assnI
thf(fact_500_in__range__subset,axiom,
! [As: set @ nat,As5: set @ nat,H2: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As @ As5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As5 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% in_range_subset
thf(fact_501_trancl__sub__insert__trancl,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R4 ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R4 ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_502_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
( ( wand_raw @ P @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H5: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( 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 @ As6 ) ) )
=> ( Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As @ As6 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_503_wand__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H5: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_504_wand__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H7: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As2 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As2 @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_505_wand__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H6: heap_ext @ product_unit,As4: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As4 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As2 @ As4 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_506_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R4 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_507_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_508_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_509_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_510_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_511_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_512_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_513_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_514_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_515_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_516_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_517_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_518_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_519_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_520_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_521_trans__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( trans @ A @ S2 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% trans_Int
thf(fact_522_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_523_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P5: product_prod @ C @ D] : ( insert2 @ D @ ( product_snd @ C @ D @ P5 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_524_snds_Ocases,axiom,
! [B: $tType,A: $tType,A4: B,P3: product_prod @ A @ B] :
( ( member @ B @ A4 @ ( basic_snds @ A @ B @ P3 ) )
=> ( A4
= ( product_snd @ A @ B @ P3 ) ) ) ).
% snds.cases
thf(fact_525_snds_Osimps,axiom,
! [B: $tType,A: $tType,A4: B,P3: product_prod @ A @ B] :
( ( member @ B @ A4 @ ( basic_snds @ A @ B @ P3 ) )
= ( A4
= ( product_snd @ A @ B @ P3 ) ) ) ).
% snds.simps
thf(fact_526_snds_Ointros,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] : ( member @ B @ ( product_snd @ A @ B @ P3 ) @ ( basic_snds @ A @ B @ P3 ) ) ).
% snds.intros
thf(fact_527_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B5 ) ) ).
% less_by_empty
thf(fact_528_wand__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H5: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_529_wand__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H7: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As2 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As2 @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_530_wand__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H6: heap_ext @ product_unit,As4: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As4 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As4 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As2 @ As4 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_531_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_532_less__eq__assn__def,axiom,
( ( ord_less_eq @ assn )
= ( ^ [A8: assn,B6: assn] :
( A8
= ( inf_inf @ assn @ A8 @ B6 ) ) ) ) ).
% less_eq_assn_def
thf(fact_533_times__assn__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) )
=> ? [As13: set @ nat,As24: set @ nat] :
( ( As2
= ( 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 ) @ H @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_534_times__assn__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) )
=> ~ ? [As12: set @ nat,As23: set @ nat] :
( ( As2
= ( 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 ) @ H @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_535_times__assn__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( Y
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As2
= ( 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 ) @ H @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) @ H @ As2 ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_536_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_537_Field__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ).
% Field_insert
thf(fact_538_dom__ran__disj__comp,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R4 ) @ ( range2 @ A @ A @ R4 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( relcomp @ A @ A @ A @ R4 @ R4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% dom_ran_disj_comp
thf(fact_539_proj__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X )
= ( equiv_proj @ A @ A @ R2 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_540_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A4 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_541_equiv__class__nondisjoint,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_542_finite__reachable__advance,axiom,
! [A: $tType,E3: set @ ( product_prod @ A @ A ),V0: A,V: A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_543_finite__Field__eq__finite,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ ( field2 @ A @ R4 ) )
= ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 ) ) ).
% finite_Field_eq_finite
thf(fact_544_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R4: set @ ( product_prod @ A @ C )] :
( ( relcomp @ A @ C @ B @ R4 @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty2
thf(fact_545_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty1
thf(fact_546_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ C ),T3: set @ ( product_prod @ A @ C ),R4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ C ) ) @ S @ T3 ) @ R4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ S @ R4 ) @ ( relcomp @ A @ C @ B @ T3 @ R4 ) ) ) ).
% relcomp_distrib2
thf(fact_547_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R4: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B ),T3: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ R4 @ ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ S @ T3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ R4 @ S ) @ ( relcomp @ A @ C @ B @ R4 @ T3 ) ) ) ).
% relcomp_distrib
thf(fact_548_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_549_Field__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) )
= ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S2 ) ) ) ).
% Field_Un
thf(fact_550_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R4: set @ ( product_prod @ A @ D ),S: set @ ( product_prod @ D @ C ),T3: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R4 @ S ) @ T3 )
= ( relcomp @ A @ D @ B @ R4 @ ( relcomp @ D @ C @ B @ S @ T3 ) ) ) ).
% O_assoc
thf(fact_551_finite__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( finite_finite2 @ A @ ( field2 @ A @ R2 ) ) ) ).
% finite_Field
thf(fact_552_finite__Image,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),A3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R4 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R4 @ A3 ) ) ) ).
% finite_Image
thf(fact_553_finite__Domain,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ A @ ( domain @ A @ B @ R2 ) ) ) ).
% finite_Domain
thf(fact_554_finite__Range,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ B @ ( range2 @ A @ B @ R2 ) ) ) ).
% finite_Range
thf(fact_555_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A4: A,C2: B,R2: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ C2 ) @ ( relcomp @ A @ C @ B @ R2 @ S2 ) )
=> ~ ! [B2: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A4 @ B2 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B2 @ C2 ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_556_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S2 ) )
=> ~ ! [X3: A,Y2: C,Z4: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z4 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z4 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_557_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B ),C2: C,S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C2 ) @ S2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A4 @ C2 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_558_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R2: 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 @ R2 @ S2 ) )
= ( ? [A8: A,B6: B,C5: C] :
( ( A1 = A8 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B6 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B6 @ C5 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_559_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R2: 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 @ R2 @ S2 ) )
=> ~ ! [B2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B2 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_560_FieldI2,axiom,
! [A: $tType,I2: A,J2: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R4 )
=> ( member @ A @ J2 @ ( field2 @ A @ R4 ) ) ) ).
% FieldI2
thf(fact_561_FieldI1,axiom,
! [A: $tType,I2: A,J2: A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R4 )
=> ( member @ A @ I2 @ ( field2 @ A @ R4 ) ) ) ).
% FieldI1
thf(fact_562_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R7: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C ),S2: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R7 @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S3 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R7 @ S3 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% relcomp_mono
thf(fact_563_relcomp__Image,axiom,
! [A: $tType,C: $tType,B: $tType,X4: set @ ( product_prod @ B @ C ),Y5: set @ ( product_prod @ C @ A ),Z6: set @ B] :
( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X4 @ Y5 ) @ Z6 )
= ( image @ C @ A @ Y5 @ ( image @ B @ C @ X4 @ Z6 ) ) ) ).
% relcomp_Image
thf(fact_564_trans__O__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ R2 ) @ R2 ) ) ).
% trans_O_subset
thf(fact_565_mono__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 )
=> ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S2 ) ) ) ).
% mono_Field
thf(fact_566_union__comp__emptyL,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A3 @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B5 @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B5 ) @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_567_union__comp__emptyR,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A3 @ B5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A3 @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A3 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B5 @ C6 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_568_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B5: set @ B,C6: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ A3 @ B5 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C6 @ A3 )
=> ( finite_finite2 @ A @ ( image @ B @ A @ C6 @ B5 ) ) ) ) ).
% finite_Image_subset
thf(fact_569_equiv__class__self,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ A @ A4 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_570_Field__def,axiom,
! [A: $tType] :
( ( field2 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) ) ) ) ).
% Field_def
thf(fact_571_equiv__class__eq__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
= ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A3 )
& ( member @ A @ Y @ A3 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_572_eq__equiv__class__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_573_equiv__class__eq,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_574_eq__equiv__class,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A,A3: set @ A] :
( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ B3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_575_rtrancl__Image__in__Field,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),V2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ V2 ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R4 ) @ V2 ) ) ).
% rtrancl_Image_in_Field
thf(fact_576_refines__equiv__class__eq2,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S )
=> ( ( equiv_equiv @ A @ A3 @ R4 )
=> ( ( equiv_equiv @ A @ A3 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R4 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_577_refines__equiv__class__eq,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S )
=> ( ( equiv_equiv @ A @ A3 @ R4 )
=> ( ( equiv_equiv @ A @ A3 @ S )
=> ( ( image @ A @ A @ R4 @ ( 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_578_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A4 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_579_finite__reachable__restrictedI,axiom,
! [A: $tType,Q2: set @ A,I: set @ A,E3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ Q2 )
=> ( ( ord_less_eq @ ( set @ A ) @ I @ Q2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ E3 ) @ Q2 )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E3 ) @ I ) ) ) ) ) ).
% finite_reachable_restrictedI
thf(fact_580_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B5: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ A3 ) @ ( image @ A @ A @ R2 @ B5 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ B5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_581_equiv__class__subset,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_582_subset__equiv__class,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B3: A,A4: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_583_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ F5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A3 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_584_finite__insert,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
thf(fact_585_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ 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_586_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_587_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S5: set @ B] :
( ( finite_finite2 @ B @ S5 )
=> ( ! [Y6: B] :
( ( member @ B @ Y6 @ S5 )
=> ( ord_less_eq @ A @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert2 @ B @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_588_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ! [A11: set @ A] :
( ~ ( finite_finite2 @ A @ A11 )
=> ( P @ A11 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_589_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ 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_590_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_591_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
( ( A8
= ( bot_bot @ ( set @ A ) ) )
| ? [A5: set @ A,B6: A] :
( ( A8
= ( insert2 @ A @ B6 @ A5 ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_592_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_593_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_594_finite_OinsertI,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( insert2 @ A @ A4 @ A3 ) ) ) ).
% finite.insertI
thf(fact_595_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ 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_596_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ 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_597_finite_Ocases,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A11: set @ A] :
( ? [A6: A] :
( A4
= ( insert2 @ A @ A6 @ A11 ) )
=> ~ ( finite_finite2 @ A @ A11 ) ) ) ) ).
% finite.cases
thf(fact_598_min__ext__compat,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R4 @ S ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R4 ) @ ( 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 @ R4 ) ) ) ).
% min_ext_compat
thf(fact_599_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_600_Total__subset__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) )
=> ( ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A6: A] :
( R2
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_601_finite__transitivity__chain,axiom,
! [A: $tType,A3: set @ A,R4: A > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X3: A] :
~ ( R4 @ X3 @ X3 )
=> ( ! [X3: A,Y2: A,Z4: A] :
( ( R4 @ X3 @ Y2 )
=> ( ( R4 @ Y2 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A3 )
& ( R4 @ X3 @ Y6 ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_602_disjnt__equiv__class,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_603_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_604_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_605_in__quotient__imp__in__rel,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_606_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_607_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E3 ) )
=> ( ~ ( member @ A @ X @ R4 )
=> ( ~ ( member @ A @ Y @ R4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ R4 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E3 @ R4 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_608_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_609_IdI,axiom,
! [A: $tType,A4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_610_quotient__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_611_quotient__is__empty,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A3 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_612_quotient__is__empty2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A3 @ R2 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_613_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_614_disjnt__insert1,axiom,
! [A: $tType,A4: A,X4: set @ A,Y5: set @ A] :
( ( disjnt @ A @ ( insert2 @ A @ A4 @ X4 ) @ Y5 )
= ( ~ ( member @ A @ A4 @ Y5 )
& ( disjnt @ A @ X4 @ Y5 ) ) ) ).
% disjnt_insert1
thf(fact_615_disjnt__insert2,axiom,
! [A: $tType,Y5: set @ A,A4: A,X4: set @ A] :
( ( disjnt @ A @ Y5 @ ( insert2 @ A @ A4 @ X4 ) )
= ( ~ ( member @ A @ A4 @ Y5 )
& ( disjnt @ A @ Y5 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_616_rel__restrict__empty,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R4 @ ( bot_bot @ ( set @ A ) ) )
= R4 ) ).
% rel_restrict_empty
thf(fact_617_disjnt__Un1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( ( disjnt @ A @ A3 @ C6 )
& ( disjnt @ A @ B5 @ C6 ) ) ) ).
% disjnt_Un1
thf(fact_618_disjnt__Un2,axiom,
! [A: $tType,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( disjnt @ A @ C6 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ( disjnt @ A @ C6 @ A3 )
& ( disjnt @ A @ C6 @ B5 ) ) ) ).
% disjnt_Un2
thf(fact_619_Image__Id,axiom,
! [A: $tType,A3: set @ A] :
( ( image @ A @ A @ ( id2 @ A ) @ A3 )
= A3 ) ).
% Image_Id
thf(fact_620_Id__O__R,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ A @ B @ ( id2 @ A ) @ R4 )
= R4 ) ).
% Id_O_R
thf(fact_621_R__O__Id,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ B @ B @ R4 @ ( id2 @ B ) )
= R4 ) ).
% R_O_Id
thf(fact_622_bijective__Id,axiom,
! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% bijective_Id
thf(fact_623_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_624_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_625_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_626_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A5: set @ A,B8: set @ A] :
! [X2: A] :
~ ( ( member @ A @ X2 @ A5 )
& ( member @ A @ X2 @ B8 ) ) ) ) ).
% disjnt_iff
thf(fact_627_disjnt__sym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( disjnt @ A @ A3 @ B5 )
=> ( disjnt @ A @ B5 @ A3 ) ) ).
% disjnt_sym
thf(fact_628_IdE,axiom,
! [A: $tType,P3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( id2 @ A ) )
=> ~ ! [X3: A] :
( P3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_629_disjnt__empty1,axiom,
! [A: $tType,A3: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% disjnt_empty1
thf(fact_630_disjnt__empty2,axiom,
! [A: $tType,A3: set @ A] : ( disjnt @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_631_disjnt__subset1,axiom,
! [A: $tType,X4: set @ A,Y5: set @ A,Z6: set @ A] :
( ( disjnt @ A @ X4 @ Y5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z6 @ X4 )
=> ( disjnt @ A @ Z6 @ Y5 ) ) ) ).
% disjnt_subset1
thf(fact_632_disjnt__subset2,axiom,
! [A: $tType,X4: set @ A,Y5: set @ A,Z6: set @ A] :
( ( disjnt @ A @ X4 @ Y5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z6 @ Y5 )
=> ( disjnt @ A @ X4 @ Z6 ) ) ) ).
% disjnt_subset2
thf(fact_633_disjnt__insert,axiom,
! [A: $tType,X: A,N: set @ A,M2: set @ A] :
( ~ ( member @ A @ X @ N )
=> ( ( disjnt @ A @ M2 @ N )
=> ( disjnt @ A @ ( insert2 @ A @ X @ M2 ) @ N ) ) ) ).
% disjnt_insert
thf(fact_634_trans__Id,axiom,
! [A: $tType] : ( trans @ A @ ( id2 @ A ) ) ).
% trans_Id
thf(fact_635_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A3 @ R4 ) )
=> ~ ( member @ A @ Y @ R4 ) ) ).
% rel_restrict_notR(2)
thf(fact_636_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A3 @ R4 ) )
=> ~ ( member @ A @ X @ R4 ) ) ).
% rel_restrict_notR(1)
thf(fact_637_rel__restrictI,axiom,
! [A: $tType,X: A,R4: set @ A,Y: A,E3: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ R4 )
=> ( ~ ( member @ A @ Y @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E3 @ R4 ) ) ) ) ) ).
% rel_restrictI
thf(fact_638_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E3 @ R4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E3 ) ) ).
% rel_restrict_lift
thf(fact_639_rel__restrict__union,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A,B5: set @ A] :
( ( rel_restrict @ A @ R4 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R4 @ A3 ) @ B5 ) ) ).
% rel_restrict_union
thf(fact_640_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id2 @ A ) ) ).
% antisym_Id
thf(fact_641_rel__restrict__mono,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A3 @ R4 ) @ ( rel_restrict @ A @ B5 @ R4 ) ) ) ).
% rel_restrict_mono
thf(fact_642_rel__restrict__sub,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R4 @ A3 ) @ R4 ) ).
% rel_restrict_sub
thf(fact_643_finite__rel__restrict,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R4 @ A3 ) ) ) ).
% finite_rel_restrict
thf(fact_644_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ 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_645_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,C6: A > ( set @ B ),B5: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ C6 ) @ ( product_Sigma @ A @ B @ B5 @ C6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
=> ( ( C6 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A3 @ B5 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_646_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ B8 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_647_quotient__eqI,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,Y5: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ A @ X @ X4 )
=> ( ( member @ A @ Y @ Y5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( X4 = Y5 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_648_quotient__eq__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,Y5: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ A @ X @ X4 )
=> ( ( member @ A @ Y @ Y5 )
=> ( ( X4 = Y5 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_649_in__quotient__imp__closed,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ A @ X @ X4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_650_in__quotient__imp__non__empty,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( X4
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_651_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W: A,E3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E3 @ R4 ) ) )
=> ~ ( member @ A @ W @ R4 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_652_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W: A,E3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E3 @ R4 ) ) )
=> ~ ( member @ A @ V @ R4 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_653_rel__restrict__trancl__mem,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A3 @ R4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A3 ) @ R4 ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_654_rel__restrict__mono2,axiom,
! [A: $tType,R4: set @ A,S: set @ A,A3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R4 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A3 @ S ) @ ( rel_restrict @ A @ A3 @ R4 ) ) ) ).
% rel_restrict_mono2
thf(fact_655_rel__restrict__trancl__sub,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),R4: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A3 @ R4 ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A3 ) @ R4 ) ) ).
% rel_restrict_trancl_sub
thf(fact_656_singleton__quotient,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_657_quotientI,axiom,
! [A: $tType,X: A,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A3 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A3 @ R2 ) ) ) ).
% quotientI
thf(fact_658_quotientE,axiom,
! [A: $tType,X4: set @ A,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ~ ! [X3: A] :
( ( X4
= ( image @ A @ A @ R2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X3 @ A3 ) ) ) ).
% quotientE
thf(fact_659_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
=> ! [A12: A] :
( ( member @ A @ A12 @ A3 )
=> ( ord_less_eq @ A @ X @ A12 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_660_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_661_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X )
=> ! [A12: A] :
( ( member @ A @ A12 @ A3 )
=> ( ord_less_eq @ A @ A12 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_662_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_663_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_664_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_665_quotient__disj,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,Y5: set @ A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( X4 = Y5 )
| ( ( inf_inf @ ( set @ A ) @ X4 @ Y5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_666_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ A3 )
=> ( ( transitive_rtrancl @ A @ A3 )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ ( id2 @ A ) ) ) ) ).
% trans_rtrancl_eq_reflcl
thf(fact_667_rel__restrict__Int__empty,axiom,
! [A: $tType,A3: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ ( field2 @ A @ R4 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R4 @ A3 )
= R4 ) ) ).
% rel_restrict_Int_empty
thf(fact_668_eq__equiv__class__iff2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_669_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_670_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_671_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_672_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B5 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_673_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_674_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_675_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_676_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_677_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_678_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_679_refl__on__reflcl__Image,axiom,
! [A: $tType,B5: set @ A,A3: set @ ( product_prod @ A @ A ),C6: set @ A] :
( ( refl_on @ A @ B5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C6 @ B5 )
=> ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ ( id2 @ A ) ) @ C6 )
= ( image @ A @ A @ A3 @ C6 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_680_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_681_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ R4 ) @ R4 )
=> ( ~ ( member @ A @ V @ R4 )
=> ~ ( ~ ( member @ A @ U @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E3 @ R4 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_682_max__ext__compat,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R4 @ S ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R4 ) @ ( 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 @ R4 ) ) ) ).
% max_ext_compat
thf(fact_683_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_684_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_685_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_686_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_687_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_688_acyclic__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R2 ) )
= ( ( transitive_acyclic @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_689_irrefl__tranclI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R2 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_690_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F2 @ A3 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_691_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_692_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,X: B,A3: set @ B] :
( ( B3
= ( F2 @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_693_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_idemp
thf(fact_694_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_695_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% DiffI
thf(fact_696_converse__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_697_converse__inject,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( ( converse @ B @ A @ R2 )
= ( converse @ B @ A @ S2 ) )
= ( R2 = S2 ) ) ).
% converse_inject
thf(fact_698_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_699_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_700_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_701_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( insert2 @ B @ ( F2 @ X ) @ ( image2 @ A @ B @ F2 @ A3 ) )
= ( image2 @ A @ B @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_702_image__insert,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: B,B5: set @ B] :
( ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A4 @ B5 ) )
= ( insert2 @ A @ ( F2 @ A4 ) @ ( image2 @ B @ A @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_703_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_704_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_705_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_706_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_707_insert__Diff1,axiom,
! [A: $tType,X: A,B5: set @ A,A3: set @ A] :
( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_708_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_709_Un__Diff__cancel2,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ B5 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_710_Un__Diff__cancel,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_Diff_cancel
thf(fact_711_converse__iff,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( converse @ B @ A @ R2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ R2 ) ) ).
% converse_iff
thf(fact_712_Field__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( converse @ A @ A @ R2 ) )
= ( field2 @ A @ R2 ) ) ).
% Field_converse
thf(fact_713_converse__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S2 ) ) ).
% converse_mono
thf(fact_714_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_715_trans__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ ( converse @ A @ A @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% trans_converse
thf(fact_716_converse__Id,axiom,
! [A: $tType] :
( ( converse @ A @ A @ ( id2 @ A ) )
= ( id2 @ A ) ) ).
% converse_Id
thf(fact_717_finite__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ B @ A ) @ R2 ) ) ).
% finite_converse
thf(fact_718_refl__on__converse,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
= ( refl_on @ A @ A3 @ R2 ) ) ).
% refl_on_converse
thf(fact_719_total__on__converse,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
= ( total_on @ A @ A3 @ R2 ) ) ).
% total_on_converse
thf(fact_720_acyclic__empty,axiom,
! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% acyclic_empty
thf(fact_721_total__on__diff__Id,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A3 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( total_on @ A @ A3 @ R2 ) ) ).
% total_on_diff_Id
thf(fact_722_antisym__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( converse @ A @ A @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisym_converse
thf(fact_723_converse__Id__on,axiom,
! [A: $tType,A3: set @ A] :
( ( converse @ A @ A @ ( id_on @ A @ A3 ) )
= ( id_on @ A @ A3 ) ) ).
% converse_Id_on
thf(fact_724_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_725_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_726_finite__Diff__insert,axiom,
! [A: $tType,A3: set @ A,A4: A,B5: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_727_Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_728_below__Id__inv,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R4 ) @ ( id2 @ A ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ).
% below_Id_inv
thf(fact_729_Domain__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( domain @ A @ B @ ( converse @ B @ A @ R2 ) )
= ( range2 @ B @ A @ R2 ) ) ).
% Domain_converse
thf(fact_730_Range__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( range2 @ B @ A @ ( converse @ A @ B @ R2 ) )
= ( domain @ A @ B @ R2 ) ) ).
% Range_converse
thf(fact_731_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A3 ) @ ( image2 @ B @ A @ F2 @ B5 ) ) @ ( image2 @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_732_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B3: B,F2: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B3
= ( F2 @ X ) )
=> ( member @ B @ B3 @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_733_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( P @ X3 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A3 )
=> ( P @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_734_image__cong,axiom,
! [B: $tType,A: $tType,M2: set @ A,N: set @ A,F2: A > B,G: A > B] :
( ( M2 = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image2 @ A @ B @ F2 @ M2 )
= ( image2 @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_735_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image2 @ B @ A @ F2 @ A3 ) )
& ( P @ X6 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_736_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Z2 @ ( image2 @ B @ A @ F2 @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( Z2
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_737_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F2 @ X ) @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_738_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( member @ A @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_739_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_740_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_741_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_742_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( A4 = B3 )
= ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_743_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I: set @ A,J: set @ A,C6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I @ J ) @ C6 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C6 ) @ ( product_Sigma @ A @ B @ J @ C6 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_744_in__image__insert__iff,axiom,
! [A: $tType,B5: set @ ( set @ A ),X: A,A3: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B5 )
=> ~ ( member @ A @ X @ C7 ) )
=> ( ( member @ ( set @ A ) @ A3 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X ) @ B5 ) )
= ( ( member @ A @ X @ A3 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_745_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_746_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
= ( ? [A8: A,B6: B] :
( ( A1 = B6 )
& ( A22 = A8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_747_converseD,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( converse @ B @ A @ R2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ R2 ) ) ).
% converseD
thf(fact_748_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
=> ~ ! [X3: B,Y2: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y2 @ X3 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_749_converseI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% converseI
thf(fact_750_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B5: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ B @ A3 ) @ ( domain @ A @ B @ B5 ) ) @ ( domain @ A @ B @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ B5 ) ) ) ).
% Domain_Diff_subset
thf(fact_751_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B5: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ B @ A @ A3 ) @ ( range2 @ B @ A @ B5 ) ) @ ( range2 @ B @ A @ ( minus_minus @ ( set @ ( product_prod @ B @ A ) ) @ A3 @ B5 ) ) ) ).
% Range_Diff_subset
thf(fact_752_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_753_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_754_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_755_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_756_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F2 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B5
= ( image2 @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_757_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A3 ) @ B5 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( member @ A @ ( F2 @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_758_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
=> ( B5
!= ( image2 @ B @ A @ F2 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_759_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_760_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A3 ) @ ( image2 @ A @ B @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_761_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S2 ) ) ).
% converse_subset_swap
thf(fact_762_image__Un,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B5: set @ B] :
( ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A3 ) @ ( image2 @ B @ A @ F2 @ B5 ) ) ) ).
% image_Un
thf(fact_763_double__diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C6 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_764_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ A3 ) ).
% Diff_subset
thf(fact_765_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C6: set @ A,D4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C6 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_766_insert__Diff__if,axiom,
! [A: $tType,X: A,B5: set @ A,A3: set @ A] :
( ( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ B5 )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_767_converse__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ B @ C ),S2: set @ ( product_prod @ C @ A )] :
( ( converse @ B @ A @ ( relcomp @ B @ C @ A @ R2 @ S2 ) )
= ( relcomp @ A @ C @ B @ ( converse @ C @ A @ S2 ) @ ( converse @ B @ C @ R2 ) ) ) ).
% converse_relcomp
thf(fact_768_converse__Int,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S2 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S2 ) ) ) ).
% converse_Int
thf(fact_769_Diff__Int__distrib2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C6 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Diff_Int_distrib2
thf(fact_770_Diff__Int__distrib,axiom,
! [A: $tType,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ C6 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C6 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C6 @ B5 ) ) ) ).
% Diff_Int_distrib
thf(fact_771_Diff__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_Diff_Int
thf(fact_772_Diff__Int2,axiom,
! [A: $tType,A3: set @ A,C6: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C6 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C6 ) @ B5 ) ) ).
% Diff_Int2
thf(fact_773_Int__Diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Int_Diff
thf(fact_774_converse__Un,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S2 ) ) ) ).
% converse_Un
thf(fact_775_set__diff__diff__left,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% set_diff_diff_left
thf(fact_776_Un__Diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ C6 ) @ ( minus_minus @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Un_Diff
thf(fact_777_vimage__Diff,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B5: set @ B] :
( ( vimage @ A @ B @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B5 ) ) ) ).
% vimage_Diff
thf(fact_778_cyclic__subset,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ R4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S )
=> ~ ( transitive_acyclic @ A @ S ) ) ) ).
% cyclic_subset
thf(fact_779_acyclic__union_I2_J,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B5 ) )
=> ( transitive_acyclic @ A @ B5 ) ) ).
% acyclic_union(2)
thf(fact_780_acyclic__union_I1_J,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B5 ) )
=> ( transitive_acyclic @ A @ A3 ) ) ).
% acyclic_union(1)
thf(fact_781_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_782_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_783_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_784_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_785_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_786_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_787_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A3 ) @ ( image2 @ B @ A @ F2 @ B5 ) ) ) ).
% image_Int_subset
thf(fact_788_subset__minus__empty,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_789_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A3 ) @ B5 )
= ( ord_less_eq @ ( set @ B ) @ A3 @ ( vimage @ B @ A @ F2 @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_790_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_791_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A4: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_792_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_793_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A4: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_794_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_795_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_796_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X4: set @ A] :
( ~ ( member @ A @ X @ X4 )
=> ( ( minus_minus @ ( set @ A ) @ X4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X4 ) ) ).
% set_minus_singleton_eq
thf(fact_797_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert2 @ A @ X @ C6 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ C6 ) )
& ~ ( member @ A @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_798_Diff__triv,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= A3 ) ) ).
% Diff_triv
thf(fact_799_Int__Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_800_disjoint__alt__simp1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_801_disjoint__alt__simp2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
!= A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_802_Diff__subset__conv,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ C6 )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% Diff_subset_conv
thf(fact_803_Diff__partition,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_804_Un__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_805_Int__Diff__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_806_Diff__Int,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C6 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% Diff_Int
thf(fact_807_Diff__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% Diff_Un
thf(fact_808_Range__snd,axiom,
! [A: $tType,B: $tType] :
( ( range2 @ B @ A )
= ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% Range_snd
thf(fact_809_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ B @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R4 )
= ( range2 @ B @ A @ R4 ) ) ).
% snd_eq_Range
thf(fact_810_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( F2 @ Y2 )
= ( F2 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F2 @ A3 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_811_cyclicE,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ G )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ G ) ) ) ).
% cyclicE
thf(fact_812_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% acyclic_def
thf(fact_813_acyclicI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( transitive_acyclic @ A @ R2 ) ) ).
% acyclicI
thf(fact_814_snd__image__mp,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ B @ A ),B5: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A3 ) @ B5 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 )
=> ( member @ A @ Y @ B5 ) ) ) ).
% snd_image_mp
thf(fact_815_snd__in__Field,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_snd @ A @ A ) @ R4 ) @ ( field2 @ A @ R4 ) ) ).
% snd_in_Field
thf(fact_816_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A3 @ B5 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A3 ) ) ).
% Image_subset_snd_image
thf(fact_817_infinite__remove,axiom,
! [A: $tType,S: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_818_infinite__coinduct,axiom,
! [A: $tType,X4: ( set @ A ) > $o,A3: set @ A] :
( ( X4 @ A3 )
=> ( ! [A11: set @ A] :
( ( X4 @ A11 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A11 )
& ( ( X4 @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_819_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: A,A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( member @ A @ A6 @ A11 )
=> ( ( P @ A11 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_820_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_821_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_822_Field__rel__restrict,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R4 @ A3 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R4 ) @ A3 ) ) ).
% Field_rel_restrict
thf(fact_823_Domain__rel__restrict,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ A @ ( rel_restrict @ A @ R4 @ A3 ) ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R4 ) @ A3 ) ) ).
% Domain_rel_restrict
thf(fact_824_Range__rel__restrict,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ ( rel_restrict @ A @ R4 @ A3 ) ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R4 ) @ A3 ) ) ).
% Range_rel_restrict
thf(fact_825_trans__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( trans @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% trans_diff_Id
thf(fact_826_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_827_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F2 @ A3 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_828_acyclic__insert__cyclic,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( transitive_acyclic @ A @ G )
=> ( ~ ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ G ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ G ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_829_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N: set @ A] :
( ! [X3: A,Y2: A] :
( ( H2 @ ( inf_inf @ A @ X3 @ Y2 ) )
= ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H2 @ N ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_830_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N: set @ A] :
( ! [X3: A,Y2: A] :
( ( H2 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H2 @ N ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_831_finite__remove__induct,axiom,
! [A: $tType,B5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( A11
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A11 @ B5 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A11 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A11 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_832_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B5: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B5 )
=> ( P @ B5 ) )
=> ( ! [A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ( A11
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A11 @ B5 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A11 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A11 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A11 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_833_Linear__order__in__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_834_max__ext_Omax__extI,axiom,
! [A: $tType,X4: set @ A,Y5: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X4 )
=> ( ( finite_finite2 @ A @ Y5 )
=> ( ( Y5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R4 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y5 ) @ ( max_ext @ A @ R4 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_835_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R4 ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A1 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R4 ) ) ) ) ) ).
% max_ext.simps
thf(fact_836_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R4 ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X6: A] :
( ( member @ A @ X6 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_837_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q2: A > $o] :
( ! [X3: A,K3: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ( ! [X3: A,K3: A] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D4 ) ) )
| ( Q2 @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_838_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q2: A > $o] :
( ! [X3: A,K3: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ( ! [X3: A,K3: A] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D4 ) ) )
& ( Q2 @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_839_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_840_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_841_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( times_times @ A @ ( minus_minus @ A @ B3 @ C2 ) @ A4 )
= ( minus_minus @ A @ ( times_times @ A @ B3 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) ) ) ) ).
% left_diff_distrib'
thf(fact_842_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_843_in__lex__prod,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A7: A,B4: B,R2: 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 @ B4 ) ) @ ( lex_prod @ A @ B @ R2 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A7 ) @ R2 )
| ( ( A4 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_844_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( ! [A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R2 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_845_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_846_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) )
=> ( Phi @ A4 @ B3 ) )
=> ( ( ( A4 = B3 )
=> ( Phi @ A4 @ B3 ) )
=> ( Phi @ A4 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_847_wf__max,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( converse @ A @ A @ R4 ) )
=> ( ( R4
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M3: A] :
~ ( member @ A @ M3 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R4 ) @ ( domain @ A @ A @ R4 ) ) ) ) ) ).
% wf_max
thf(fact_848_wf__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( wf @ A @ S2 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R2 ) @ ( range2 @ A @ A @ S2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_849_Linear__order__Well__order__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( ! [A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R2 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_850_underS__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( order_underS @ A @ R2 @ B3 ) ) ) ) ) ).
% underS_incr
thf(fact_851_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A13: set @ A,R1: set @ ( product_prod @ A @ A ),A23: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > C] :
( ( equiv_equiv @ A @ A13 @ R1 )
=> ( ( equiv_equiv @ B @ A23 @ R22 )
=> ( ! [Y2: A,Z4: A,W2: B] :
( ( member @ B @ W2 @ A23 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R1 )
=> ( ( F2 @ Y2 @ W2 )
= ( F2 @ Z4 @ W2 ) ) ) )
=> ( ! [Y2: B,Z4: B,W2: A] :
( ( member @ A @ W2 @ A13 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z4 ) @ R22 )
=> ( ( F2 @ W2 @ Y2 )
= ( F2 @ W2 @ Z4 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F2 ) ) ) ) ) ).
% congruent2I
thf(fact_852_member__remove,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A3 ) )
= ( ( member @ A @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_853_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_854_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R2 ) )
= ( ( wf @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% wf_insert
thf(fact_855_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( ( Y6 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 ) )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_856_wf__induct__rule,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( wf @ A @ R2 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% wf_induct_rule
thf(fact_857_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [Q: set @ A] :
( ? [X2: A] : ( member @ A @ X2 @ Q )
=> ? [X2: A] :
( ( member @ A @ X2 @ Q )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 )
=> ~ ( member @ A @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_858_wf__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_859_wf__not__sym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,X: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A4 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_860_wf__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_861_wf__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( wf @ A @ R2 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% wf_induct
thf(fact_862_wf__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,X: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A4 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_863_wfUNIVI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [P4: A > $o,X3: A] :
( ! [Xa2: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R2 )
=> ( P4 @ Y2 ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf @ A @ R2 ) ) ).
% wfUNIVI
thf(fact_864_wfI__min,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Q3: set @ A] :
( ( member @ A @ X3 @ Q3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Q3 )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R4 )
=> ~ ( member @ A @ Y2 @ Q3 ) ) ) )
=> ( wf @ A @ R4 ) ) ).
% wfI_min
thf(fact_865_wfE__min,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),X: A,Q2: set @ A] :
( ( wf @ A @ R4 )
=> ( ( member @ A @ X @ Q2 )
=> ~ ! [Z4: A] :
( ( member @ A @ Z4 @ Q2 )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z4 ) @ R4 )
=> ~ ( member @ A @ Y6 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_866_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [P2: A > $o] :
( ! [X2: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 )
=> ( P2 @ Y3 ) )
=> ( P2 @ X2 ) )
=> ! [X7: A] : ( P2 @ X7 ) ) ) ) ).
% wf_def
thf(fact_867_well__order__on__domain,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_well_order_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ A @ A4 @ A3 )
& ( member @ A @ B3 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_868_underS__E,axiom,
! [A: $tType,I2: A,R4: set @ ( product_prod @ A @ A ),J2: A] :
( ( member @ A @ I2 @ ( order_underS @ A @ R4 @ J2 ) )
=> ( ( I2 != J2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R4 ) ) ) ).
% underS_E
thf(fact_869_underS__I,axiom,
! [A: $tType,I2: A,J2: A,R4: set @ ( product_prod @ A @ A )] :
( ( I2 != J2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R4 )
=> ( member @ A @ I2 @ ( order_underS @ A @ R4 @ J2 ) ) ) ) ).
% underS_I
thf(fact_870_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [A: $tType,I2: A,R4: set @ ( product_prod @ A @ A ),J2: A] :
( ( member @ A @ I2 @ ( order_underS @ A @ R4 @ J2 ) )
=> ( member @ A @ I2 @ ( field2 @ A @ R4 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_871_wo__rel_OTOTALS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ! [X6: A] :
( ( member @ A @ X6 @ ( field2 @ A @ R2 ) )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Xa2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_872_wfE__min_H,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),Q2: set @ A] :
( ( wf @ A @ R4 )
=> ( ( Q2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z4: A] :
( ( member @ A @ Z4 @ Q2 )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z4 ) @ R4 )
=> ~ ( member @ A @ Y6 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_873_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_874_wf__no__loop,axiom,
! [B: $tType,R4: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R4 @ R4 )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R4 ) ) ).
% wf_no_loop
thf(fact_875_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( ( wf @ A @ ( converse @ A @ A @ R4 ) )
= ( wf @ A @ R4 ) ) ) ).
% finite_wf_eq_wf_converse
thf(fact_876_well__order__induct__imp,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( ( Y6 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 ) )
=> ( ( member @ A @ Y6 @ ( field2 @ A @ R2 ) )
=> ( P @ Y6 ) ) )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( P @ A4 ) ) ) ) ).
% well_order_induct_imp
thf(fact_877_underS__empty,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( order_underS @ A @ R2 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_878_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_879_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_880_wfI__pf,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ! [A11: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A11 @ ( image @ A @ A @ R4 @ A11 ) )
=> ( A11
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R4 ) ) ).
% wfI_pf
thf(fact_881_wfE__pf,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( wf @ A @ R4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ R4 @ A3 ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_882_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K )
=> ? [X3: B] :
( ( P @ X3 )
& ! [Y6: B] :
( ( P @ Y6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y6 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_883_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A5: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R5 ) )
& ( A5
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_884_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( Phi @ A4 @ B3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 )
=> ( Phi @ A4 @ B3 ) )
=> ( Phi @ A4 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_885_wf__no__path,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R4 ) @ ( range2 @ A @ A @ R4 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ R4 ) ) ).
% wf_no_path
thf(fact_886_wf__min,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R4 )
=> ( ( R4
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M3: A] :
~ ( member @ A @ M3 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R4 ) @ ( range2 @ A @ A @ R4 ) ) ) ) ) ).
% wf_min
thf(fact_887_underS__incl__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( order_underS @ A @ R2 @ B3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_888_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F2: A > A > B] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ! [Y2: A,Z4: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( member @ A @ Z4 @ A3 )
=> ( ( F2 @ Y2 @ Z4 )
= ( F2 @ Z4 @ Y2 ) ) ) )
=> ( ! [Y2: A,Z4: A,W2: A] :
( ( member @ A @ W2 @ A3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( ( F2 @ W2 @ Y2 )
= ( F2 @ W2 @ Z4 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F2 ) ) ) ) ).
% congruent2_commuteI
thf(fact_889_brk__rel__wf,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R4 )
=> ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R4 ) ) ) ).
% brk_rel_wf
thf(fact_890_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R2 @ B5 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_891_Refl__under__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( order_under @ A @ R2 @ A4 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_892_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_893_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_894_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A4 @ B5 )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ B5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B2 ) @ R2 ) )
=> ( A4
= ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_895_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ B5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) @ B3 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_896_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_897_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R4 )
=> ( ! [F: A > B,G3: A > B,X3: A,R: B] :
( ! [Z7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z7 @ X3 ) @ R4 )
=> ( ( F @ Z7 )
= ( G3 @ Z7 ) ) )
=> ( ( P @ F @ X3 @ R )
= ( P @ G3 @ X3 @ R ) ) )
=> ( ! [X3: A,F: A > B] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R4 )
=> ( P @ F @ Y6 @ ( F @ Y6 ) ) )
=> ? [X_12: B] : ( P @ F @ X3 @ X_12 ) )
=> ? [F: A > B] :
! [X6: A] : ( P @ F @ X6 @ ( F @ X6 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_898_wo__rel_Omax2__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= B3 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= A4 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_899_under__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A4 ) @ ( order_under @ A @ R2 @ B3 ) ) ) ) ).
% under_incr
thf(fact_900_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R2 @ B5 @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B5 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_901_wo__rel_OisMinim__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R2 @ A3 @ B3 )
= ( ( member @ A @ B3 @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_902_wo__rel_Omax2__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_903_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= B3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_904_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= A4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_905_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) @ ( insert2 @ A @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_906_chains__extend,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A ),Z2: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains2 @ A @ S ) )
=> ( ( member @ ( set @ A ) @ Z2 @ S )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( 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 ) ) ) ) @ C2 ) @ ( chains2 @ A @ S ) ) ) ) ) ).
% chains_extend
thf(fact_907_Zorns__po__lemma,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ ( chains @ A @ R2 ) )
=> ? [X6: A] :
( ( member @ A @ X6 @ ( field2 @ A @ R2 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X6 ) @ R2 ) ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_908_bsqr__max2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A1: A,A22: A,B1: A,B22: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% bsqr_max2
thf(fact_909_wf__Union,axiom,
! [A: $tType,R4: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R4 )
=> ( wf @ A @ X3 ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R4 )
=> ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R4 )
=> ( ( 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 ) ) @ R4 ) ) ) ) ).
% wf_Union
thf(fact_910_Pow__singleton__iff,axiom,
! [A: $tType,X4: set @ A,Y5: set @ A] :
( ( ( pow2 @ A @ X4 )
= ( insert2 @ ( set @ A ) @ Y5 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X4
= ( bot_bot @ ( set @ A ) ) )
& ( Y5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_911_Pow__empty,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_912_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,E3: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ E3 ) )
=> ( 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 ) @ E3 ) ) ) ) ).
% rtrancl_mapI
thf(fact_913_disjoint__image__subset,axiom,
! [A: $tType,A14: set @ ( set @ A ),F2: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A14 )
=> ( ! [X8: set @ A] :
( ( member @ ( set @ A ) @ X8 @ A14 )
=> ( ord_less_eq @ ( set @ A ) @ ( F2 @ X8 ) @ X8 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F2 @ A14 ) ) ) ) ).
% disjoint_image_subset
thf(fact_914_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,B2: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B2 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B2 ) ) )
=> ~ ( 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 @ B2 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_915_Pow__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% Pow_iff
thf(fact_916_PowI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B5 ) ) ) ).
% PowI
thf(fact_917_Pow__Int__eq,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B5 ) ) ) ).
% Pow_Int_eq
thf(fact_918_Field__Union,axiom,
! [A: $tType,R4: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( field2 @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A ) @ ( field2 @ A ) @ R4 ) ) ) ).
% Field_Union
thf(fact_919_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R3: A > A > $o,S6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S6 )
=> ( ( X2 != Y3 )
=> ( R3 @ X2 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_920_pairwiseI,axiom,
! [A: $tType,S: set @ A,R4: A > A > $o] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R4 @ X3 @ Y2 ) ) ) )
=> ( pairwise @ A @ R4 @ S ) ) ).
% pairwiseI
thf(fact_921_pairwiseD,axiom,
! [A: $tType,R4: A > A > $o,S: set @ A,X: A,Y: A] :
( ( pairwise @ A @ R4 @ S )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( X != Y )
=> ( R4 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_922_Pow__top,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ A3 ) ) ).
% Pow_top
thf(fact_923_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S: set @ A,A3: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( refl_on @ B @ ( A3 @ X3 ) @ ( R2 @ X3 ) ) )
=> ( refl_on @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ S ) ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_924_Pow__bottom,axiom,
! [A: $tType,B5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B5 ) ) ).
% Pow_bottom
thf(fact_925_Pow__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B5 ) ) ) ).
% Pow_mono
thf(fact_926_PowD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% PowD
thf(fact_927_Un__Pow__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B5 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% Un_Pow_subset
thf(fact_928_Pow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( pow2 @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_929_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,P: B > B > $o] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( X3 != Y2 )
=> ( ( ( F2 @ X3 )
!= ( F2 @ Y2 ) )
=> ( P @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_930_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_931_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T3: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( pairwise @ A @ P @ T3 ) ) ) ).
% pairwise_subset
thf(fact_932_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A3: set @ A,Q2: A > A > $o,B5: set @ A] :
( ( pairwise @ A @ P @ A3 )
=> ( ! [X3: A,Y2: A] :
( ( P @ X3 @ Y2 )
=> ( Q2 @ X3 @ Y2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( pairwise @ A @ Q2 @ B5 ) ) ) ) ).
% pairwise_mono
thf(fact_933_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X: A,S2: set @ A] :
( ( pairwise @ A @ R2 @ ( insert2 @ A @ X @ S2 ) )
= ( ! [Y3: A] :
( ( ( member @ A @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise @ A @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_934_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B,A4: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A3 ) ) )
=> ( ( member @ B @ A4 @ A3 )
=> ( finite_finite2 @ A @ ( B5 @ A4 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_935_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B5: set @ A] :
( ( ( image2 @ B @ A @ F2 @ A3 )
= B5 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) )
= ( pow2 @ A @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_936_Pow__insert,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( pow2 @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ A4 ) @ ( pow2 @ A @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_937_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_938_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,B2: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B2 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B2 ) ) ) ) ) ).
% pairself.elims
thf(fact_939_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_940_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( member @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_941_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_942_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_943_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A3 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) @ ( pow2 @ A @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_944_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R3: A > A > $o,S6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( minus_minus @ ( set @ A ) @ S6 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R3 @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_945_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_946_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_947_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_948_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_949_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_950_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_951_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= ( lattic5882676163264333800up_fin @ A @ X4 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_952_Union__image__insert,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A4: B,B5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( insert2 @ B @ A4 @ B5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_953_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_954_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S6: set @ ( A > $o ),X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_955_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_956_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_957_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_958_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,Z2: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X4 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_959_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A4: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ X3 @ A4 ) )
=> ( ! [Y2: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X4 )
=> ( ord_less_eq @ A @ X6 @ Y2 ) )
=> ( ord_less_eq @ A @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= A4 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_960_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: set @ B,F2: B > A,X: A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_961_Union__disjoint,axiom,
! [A: $tType,C6: set @ ( set @ A ),A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C6 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_962_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_963_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: set @ B,C2: A,F2: B > A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I )
=> ( ( F2 @ X2 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_964_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,M2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ M2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ M2 ) ) ) ) ).
% cSUP_least
thf(fact_965_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_966_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A4: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B5 ) @ A4 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B5 )
=> ( ( inf_inf @ A @ X2 @ A4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_967_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A3: B > ( set @ A ),I2: B,B5: set @ A,J: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A3 @ I2 @ B5 ) @ J ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ B ) @ J @ ( insert2 @ B @ I2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I2 @ J ) @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_968_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,B5: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ B5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ B5 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_969_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_970_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_971_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_972_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_973_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ B @ A ),A3: set @ B,B5: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R4 ) )
=> ( ( image @ B @ A @ R4 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R4 @ A3 ) @ ( image @ B @ A @ R4 @ B5 ) ) ) ) ).
% Image_Int_eq
thf(fact_974_swap__swap,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P3 ) )
= P3 ) ).
% swap_swap
thf(fact_975_image__update,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F2: A > B,N2: B] :
( ~ ( member @ A @ X @ A3 )
=> ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F2 @ X @ N2 ) @ A3 )
= ( image2 @ A @ B @ F2 @ A3 ) ) ) ).
% image_update
thf(fact_976_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_977_bdd__above__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( condit941137186595557371_above @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( condit941137186595557371_above @ A @ A3 ) ) ) ).
% bdd_above_insert
thf(fact_978_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_979_single__valuedD,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ R2 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_980_single__valuedI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y2: B,Z4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z4 ) @ R2 )
=> ( Y2 = Z4 ) ) )
=> ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedI
thf(fact_981_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
! [X2: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 )
=> ! [Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R5 )
=> ( Y3 = Z3 ) ) ) ) ) ).
% single_valued_def
thf(fact_982_single__valued__subset,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 )
=> ( ( single_valued @ A @ B @ S2 )
=> ( single_valued @ A @ B @ R2 ) ) ) ).
% single_valued_subset
thf(fact_983_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_984_single__valued__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( single_valued @ A @ B @ R2 )
=> ( ( single_valued @ B @ C @ S2 )
=> ( single_valued @ A @ C @ ( relcomp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% single_valued_relcomp
thf(fact_985_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R4 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S ) ) ) ).
% single_valued_inter1
thf(fact_986_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R4 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S @ R4 ) ) ) ).
% single_valued_inter2
thf(fact_987_single__valued__Id,axiom,
! [A: $tType] : ( single_valued @ A @ A @ ( id2 @ A ) ) ).
% single_valued_Id
thf(fact_988_single__valued__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( single_valued @ A @ A @ ( id_on @ A @ A3 ) ) ).
% single_valued_Id_on
thf(fact_989_single__valued__confluent,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( single_valued @ A @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_990_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_991_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ B5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A3 )
& ( ord_less_eq @ A @ B2 @ X6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B5 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_992_single__valued__below__Id,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) )
=> ( single_valued @ A @ A @ R4 ) ) ).
% single_valued_below_Id
thf(fact_993_bijective__alt,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R3 )
& ( single_valued @ B @ A @ ( converse @ A @ B @ R3 ) ) ) ) ) ).
% bijective_alt
thf(fact_994_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_995_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: C > A,B5: set @ C,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G @ B5 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A3 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B5 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_996_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_997_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A4: A] :
( ( condit941137186595557371_above @ A @ X4 )
=> ( ( ( X4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ X4 ) )
= A4 ) )
& ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ X4 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_998_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A4: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X4 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A4 @ X4 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ X4 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_999_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1000_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_1001_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A3 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1002_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A3 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1003_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F2: A > ( set @ B )] :
( ! [A6: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A6 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B2 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ B2 ) @ ( Ub @ A6 ) )
& ( ord_less @ ( set @ B ) @ ( F2 @ A6 ) @ ( F2 @ B2 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_1004_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ B5 )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1005_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T4: set @ A] :
( ( ord_less @ ( set @ A ) @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X6: A] :
( ( member @ A @ X6 @ ( minus_minus @ ( set @ A ) @ S @ T4 ) )
& ( P @ ( insert2 @ A @ X6 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1006_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_1007_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: set @ A,N: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
=> ( ( M2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M2 ) @ ( lattic643756798349783984er_Max @ A @ N ) ) ) ) ) ) ).
% Max_mono
thf(fact_1008_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ A] :
( ( image2 @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_1009_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_1010_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_1011_converse__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_UNIV
thf(fact_1012_Int__UNIV,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( top_top @ ( set @ A ) ) )
= ( ( A3
= ( top_top @ ( set @ A ) ) )
& ( B5
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_1013_psubsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% psubsetI
thf(fact_1014_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_1015_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_1016_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_1017_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_1018_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_1019_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_1020_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_1021_Domain__Id,axiom,
! [A: $tType] :
( ( domain @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_Id
thf(fact_1022_Range__Id,axiom,
! [A: $tType] :
( ( range2 @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Range_Id
thf(fact_1023_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1024_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1025_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_1026_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_1027_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_1028_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_1029_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_1030_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_1031_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_1032_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_1033_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_1034_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_1035_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_1036_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_1037_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P6: A > $o] :
? [X9: A] : ( P6 @ X9 ) )
= ( ^ [P2: A > $o] :
? [N4: A] :
( ( P2 @ N4 )
& ! [M4: A] :
( ( ord_less @ A @ M4 @ N4 )
=> ~ ( P2 @ M4 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_1038_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B2: A] :
( ( ord_less @ A @ A6 @ B2 )
=> ( P @ A6 @ B2 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B2: A] :
( ( P @ B2 @ A6 )
=> ( P @ A6 @ B2 ) )
=> ( P @ A4 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_1039_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_1040_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% top.extremum_strict
thf(fact_1041_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_1042_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_1043_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_1044_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_1045_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_1046_psubsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_1047_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_1048_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_1049_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% psubset_trans
thf(fact_1050_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_1051_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_1052_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_1053_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_1054_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_1055_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( A4
= ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1056_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ B @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1057_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_1058_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_1059_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1067_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_1068_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_1069_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_1070_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_1071_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_1072_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_1073_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_1074_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_1075_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_1076_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_1077_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_1078_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_1079_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B6: A] :
( ( ord_less @ A @ A8 @ B6 )
| ( A8 = B6 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_1080_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ A8 @ B6 )
& ( A8 != B6 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_1081_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_1082_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_1083_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ A8 @ B6 )
& ~ ( ord_less_eq @ A @ B6 @ A8 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_1084_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z2 @ W2 )
=> ( ( ord_less @ A @ W2 @ X )
=> ( ord_less_eq @ A @ Y @ W2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_1085_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W2: A] :
( ( ord_less @ A @ X @ W2 )
=> ( ( ord_less @ A @ W2 @ Y )
=> ( ord_less_eq @ A @ W2 @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_1086_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A8: A] :
( ( ord_less @ A @ B6 @ A8 )
| ( A8 = B6 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1087_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A8: A] :
( ( ord_less_eq @ A @ B6 @ A8 )
& ( A8 != B6 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1088_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_1089_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_1090_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A8: A] :
( ( ord_less_eq @ A @ B6 @ A8 )
& ~ ( ord_less_eq @ A @ A8 @ B6 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1091_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_1092_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_1093_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% order_le_less
thf(fact_1094_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% order_less_le
thf(fact_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1103_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_1104_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1105_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_1106_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_1107_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_1108_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_1109_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1110_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A4 @ B3 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1111_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1112_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1113_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_1114_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_1115_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_1116_psubsetE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% psubsetE
thf(fact_1117_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B8 )
& ( A5 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_1118_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1119_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% psubset_subset_trans
thf(fact_1120_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B8 )
& ~ ( ord_less_eq @ ( set @ A ) @ B8 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1121_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% subset_psubset_trans
thf(fact_1122_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B8 )
| ( A5 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1123_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X: B] : ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_1124_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,X: B] :
( ( B3
= ( F2 @ X ) )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_1125_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_1126_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_1127_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ? [B2: A] : ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1128_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert2 @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_1129_Int__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_1130_Int__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= A3 ) ).
% Int_UNIV_right
thf(fact_1131_Un__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_1132_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_1133_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep2: A > B,Abs2: B > A,F5: B > $o] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( ? [X7: B] : ( F5 @ X7 ) )
= ( ? [B6: A] : ( F5 @ ( Rep2 @ B6 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_1134_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A,S2: A] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X3: B] :
( S2
!= ( Abs2 @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_1135_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ).
% less_1_mult
thf(fact_1136_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: set @ A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X4 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X4 ) ) ) ) ).
% infinite_growing
thf(fact_1137_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ 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_1138_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Z2: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X4 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_1139_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X4: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X4 ) )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_1140_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ).
% Max_in
thf(fact_1141_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B5: set @ A,I2: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B5 )
=> ( member @ A @ ( F2 @ I2 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1142_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_1143_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R2: set @ ( product_prod @ B @ B ),F2: B > A] :
( ! [A6: B,B2: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B2 ) @ R2 )
=> ( ord_less @ A @ ( F2 @ B2 ) @ ( F2 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R2 ) ) ) ).
% acyclicI_order
thf(fact_1144_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( image2 @ A @ B @ Abs2 @ A3 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_1145_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( image2 @ B @ A @ Rep2 @ ( top_top @ ( set @ B ) ) )
= A3 ) ) ).
% type_definition.Rep_range
thf(fact_1146_refl__Id,axiom,
! [A: $tType] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( id2 @ A ) ) ).
% refl_Id
thf(fact_1147_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_1148_union__fold__insert,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ B5 @ A3 ) ) ) ).
% union_fold_insert
thf(fact_1149_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B2: A,A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A11 )
=> ( ord_less @ A @ B2 @ X6 ) )
=> ( ( P @ A11 )
=> ( P @ ( insert2 @ A @ B2 @ A11 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1150_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B2: A,A11: set @ A] :
( ( finite_finite2 @ A @ A11 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A11 )
=> ( ord_less @ A @ X6 @ B2 ) )
=> ( ( P @ A11 )
=> ( P @ ( insert2 @ A @ B2 @ A11 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1151_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,A4: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X4 ) @ A4 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X4 )
=> ( ord_less @ A @ X2 @ A4 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1152_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1153_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1154_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1155_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
=> ! [A12: A] :
( ( member @ A @ A12 @ A3 )
=> ( ord_less_eq @ A @ A12 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_1156_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1157_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A4 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ A4 @ A3 ) )
= A4 ) ) ) ) ).
% Max_insert2
thf(fact_1158_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_1159_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X4 )
= ( lattic643756798349783984er_Max @ A @ X4 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_1160_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Max_Sup
thf(fact_1161_disjoint__alt__simp3,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_1162_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_1163_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Y: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X4 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X4 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( ord_less @ A @ Y @ X2 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_1164_underS__Field3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field3
thf(fact_1165_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X6: A] :
( ( member @ A @ X6 @ S )
& ( ord_less @ B @ ( F2 @ X6 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1166_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( complete_Sup_Sup @ A @ A3 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% Sup_fold_sup
thf(fact_1167_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,A3: set @ A,B5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( finite_fold @ A @ B @ F2 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) @ B5 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_1168_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_finite2 @ 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_1169_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_finite2 @ 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_1170_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A3: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_1171_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A3: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X @ Z2 ) @ A3 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_1172_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ 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_1173_finite__subset__Union__chain,axiom,
! [A: $tType,A3: set @ A,B10: set @ ( set @ A ),A14: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B10 ) )
=> ( ( B10
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A14 @ ( ord_less @ ( set @ A ) ) @ B10 )
=> ~ ! [B7: set @ A] :
( ( member @ ( set @ A ) @ B7 @ B10 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ B7 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_1174_comp__fun__commute__on_Ofold__insert,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_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_1175_comp__fun__commute__on_Ofold__insert2,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_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X @ Z2 ) @ A3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_1176_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_1177_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_1178_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_1179_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_1180_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F2 @ G )
= Fg )
= ( ! [X2: A] :
( ( F2 @ ( G @ X2 ) )
= ( Fg @ X2 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_1181_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F2: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F2 @ X )
= ( comp @ B @ C @ A @ F2 @ Y ) ) ) ).
% comp_cong_right
thf(fact_1182_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F2: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F2 )
= ( comp @ A @ B @ C @ Y @ F2 ) ) ) ).
% comp_cong_left
thf(fact_1183_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G: C > B,X: C,H2: D > A,K: C > D] :
( ( ( F2 @ ( G @ X ) )
= ( H2 @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G @ X )
= ( comp @ D @ A @ C @ H2 @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_1184_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F3: $tType,Rep2: A > B,Abs2: B > A,S: B > ( set @ D ),M2: C > B,F2: E > D,S4: C > ( set @ E ),G: F3 > C] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( ( comp @ B @ ( set @ D ) @ C @ S @ M2 )
= ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F2 ) @ S4 ) )
=> ( ( comp @ A @ ( set @ D ) @ F3 @ ( comp @ B @ ( set @ D ) @ A @ S @ Rep2 ) @ ( comp @ C @ A @ F3 @ ( comp @ B @ A @ C @ Abs2 @ M2 ) @ G ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F3 @ ( image2 @ E @ D @ F2 ) @ ( comp @ C @ ( set @ E ) @ F3 @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1185_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_1186_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_1187_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_1188_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_1189_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > A,X4: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F2 @ X4 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ X4 )
=> ( ( comp @ A @ C @ B @ G1 @ F2 @ X3 )
= ( comp @ A @ C @ B @ G22 @ F2 @ X3 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_1190_type__copy__map__comp0,axiom,
! [F3: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep2: A > B,Abs2: B > A,M2: C > D,M1: B > D,M22: C > B,F2: D > F3,G: E > C] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( M2
= ( comp @ B @ D @ C @ M1 @ M22 ) )
=> ( ( comp @ C @ F3 @ E @ ( comp @ D @ F3 @ C @ F2 @ M2 ) @ G )
= ( comp @ A @ F3 @ E @ ( comp @ B @ F3 @ A @ ( comp @ D @ F3 @ B @ F2 @ M1 ) @ Rep2 ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1191_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,Rep2: A > B,Abs2: B > A,Rep3: C > D,Abs3: D > C,Rep4: E > F3,Abs4: F3 > E,M2: F3 > D,M1: B > D,M22: F3 > B] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep3 @ Abs3 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F3 @ Rep4 @ Abs4 @ ( top_top @ ( set @ F3 ) ) )
=> ( ( ( comp @ F3 @ C @ E @ ( comp @ D @ C @ F3 @ Abs3 @ M2 ) @ Rep4 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs3 @ M1 ) @ Rep2 ) @ ( comp @ F3 @ A @ E @ ( comp @ B @ A @ F3 @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp @ B @ D @ F3 @ M1 @ M22 )
= M2 ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1192_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A6: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A6 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B2 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ nat @ ( F2 @ B2 ) @ ( Ub @ A6 ) )
& ( ord_less @ nat @ ( F2 @ A6 ) @ ( F2 @ B2 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_1193_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_1194_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A8: assn,B6: assn] :
( ( ord_less_eq @ assn @ A8 @ B6 )
& ( A8 != B6 ) ) ) ) ).
% less_assn_def
thf(fact_1195_top__assn__def,axiom,
( ( top_top @ assn )
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_1196_Union__in__chain,axiom,
! [A: $tType,B10: set @ ( set @ A ),A14: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B10 )
=> ( ( B10
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A14 @ ( ord_less @ ( set @ A ) ) @ B10 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B10 ) @ B10 ) ) ) ) ).
% Union_in_chain
thf(fact_1197_pred__on_Ochain__extend,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o,C6: set @ A,Z2: A] :
( ( pred_chain @ A @ A3 @ P @ C6 )
=> ( ( member @ A @ Z2 @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ C6 )
=> ( 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 ) ) ) @ C6 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1198_subset__Zorn__nonempty,axiom,
! [A: $tType,A14: set @ ( set @ A )] :
( ( A14
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C8: set @ ( set @ A )] :
( ( C8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A14 @ ( ord_less @ ( set @ A ) ) @ C8 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) @ A14 ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A14 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A14 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_1199_subset_Ochain__extend,axiom,
! [A: $tType,A3: set @ ( set @ A ),C6: set @ ( set @ A ),Z2: set @ A] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( member @ ( set @ A ) @ Z2 @ A3 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C6 )
=> ( 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 ) ) ) ) @ C6 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_1200_comp__fun__commute__on_Ofold__fun__left__comm,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_finite2 @ A @ A3 )
=> ( ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X @ Z2 ) @ A3 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_1201_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_1202_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_1203_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_1204_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1205_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1206_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ( inj_on @ A @ B @ F2 @ A3 )
& ( inj_on @ A @ B @ F2 @ B5 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_1207_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1208_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: A > B,G: C > D] :
( ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F2 @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_1209_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_1210_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_1211_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_1212_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_1213_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_1214_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_1215_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_1216_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G: D > B,A4: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F2 @ G @ ( product_Pair @ C @ D @ A4 @ B3 ) )
= ( product_Pair @ A @ B @ ( F2 @ A4 ) @ ( G @ B3 ) ) ) ).
% map_prod_simp
thf(fact_1217_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: A > D,G: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F2 @ G ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_1218_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: C > B,G: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F2 @ G @ X ) )
= ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_1219_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A4: A,B3: B,R4: set @ ( product_prod @ A @ B ),F2: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R4 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ A4 ) @ ( G @ B3 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F2 @ G ) @ R4 ) ) ) ).
% map_prod_imageI
thf(fact_1220_exists__leI,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ N2 )
=> ~ ( P @ N5 ) )
=> ( P @ N2 ) )
=> ? [N6: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
& ( P @ N6 ) ) ) ).
% exists_leI
thf(fact_1221_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_1222_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_1223_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_1224_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_1225_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_1226_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A8: A,B6: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B6 ) @ B6 @ A8 ) ) ) ) ).
% max_def
thf(fact_1227_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_1228_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_1229_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_1230_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F2: C > E,G: D > F3,H2: A > C,I2: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F3 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F3 @ F2 @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H2 @ I2 ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F2 @ H2 ) @ ( comp @ D @ F3 @ B @ G @ I2 ) ) ) ).
% map_prod.comp
thf(fact_1231_map__prod_Ocompositionality,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F2: C > E,G: D > F3,H2: A > C,I2: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F3 @ F2 @ G @ ( product_map_prod @ A @ C @ B @ D @ H2 @ I2 @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F2 @ H2 ) @ ( comp @ D @ F3 @ B @ G @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_1232_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: E > C,F22: A > E,G1: F3 > D,G22: B > F3] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F3 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F3 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F3 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F3 @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_1233_inj__swap,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A3 ) ).
% inj_swap
thf(fact_1234_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_1235_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_1236_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X: A,A3: set @ A,F2: A > B > B,Z2: B,Y: B] :
( ~ ( member @ A @ X @ A3 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A3 @ Y )
=> ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) @ ( F2 @ X @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_1237_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X: B,B5: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ~ ( member @ B @ X @ B5 )
=> ( ( ( insert2 @ B @ X @ B5 )
= ( image2 @ A @ B @ F2 @ A3 ) )
=> ~ ! [X10: A,A9: set @ A] :
( ~ ( member @ A @ X10 @ A9 )
=> ( ( A3
= ( insert2 @ A @ X10 @ A9 ) )
=> ( ( X
= ( F2 @ X10 ) )
=> ( B5
!= ( image2 @ A @ B @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1238_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep2: A > B,Abs2: B > A,X: C,S: B > ( set @ C ),Y: B] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X @ ( comp @ B @ ( set @ C ) @ A @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member @ C @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_1239_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F2: C > A,G: D > B,R4: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F2 @ G ) @ R4 ) )
=> ~ ! [X3: C,Y2: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F2 @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y2 ) @ R4 ) ) ) ).
% prod_fun_imageE
thf(fact_1240_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F4: A > B > B,Z3: B,A15: set @ A,A24: B] :
( ( ( A15
= ( bot_bot @ ( set @ A ) ) )
& ( A24 = Z3 ) )
| ? [X2: A,A5: set @ A,Y3: B] :
( ( A15
= ( insert2 @ A @ X2 @ A5 ) )
& ( A24
= ( F4 @ X2 @ Y3 ) )
& ~ ( member @ A @ X2 @ A5 )
& ( finite_fold_graph @ A @ B @ F4 @ Z3 @ A5 @ Y3 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_1241_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,A11: set @ A] :
( ( A1
= ( insert2 @ A @ X3 @ A11 ) )
=> ! [Y2: B] :
( ( A22
= ( F2 @ X3 @ Y2 ) )
=> ( ~ ( member @ A @ X3 @ A11 )
=> ~ ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A11 @ Y2 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_1242_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,A10: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A3 ) @ A10 ) ) )
= ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ A10 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1243_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_1244_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N: set @ A] :
( ! [X3: A,Y2: A] :
( ( H2 @ ( ord_max @ A @ X3 @ Y2 ) )
= ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H2 @ N ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_1245_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B5 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_1246_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1247_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Max.closed
thf(fact_1248_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_1249_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: A > B > B,X: A,A3: set @ A,Z2: B,V: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ S )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) @ V )
=> ( ~ ( member @ A @ X @ A3 )
=> ~ ! [Y2: B] :
( ( V
= ( F2 @ X @ Y2 ) )
=> ~ ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A3 @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_1250_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A3 ) ) ) ) ).
% Max.eq_fold
thf(fact_1251_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_1252_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_1253_total__inv__image,axiom,
! [B: $tType,A: $tType,F2: A > B,R2: set @ ( product_prod @ B @ B )] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R2 )
=> ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R2 @ F2 ) ) ) ) ).
% total_inv_image
thf(fact_1254_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_1255_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1256_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X: A,Y: A] :
( ( finite_finite2 @ 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_1257_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_finite2 @ A @ A3 )
& ( member @ A @ X @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1258_Inter__in__chain,axiom,
! [A: $tType,B10: set @ ( set @ A ),A14: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B10 )
=> ( ( B10
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A14 @ ( ord_less @ ( set @ A ) ) @ B10 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B10 ) @ B10 ) ) ) ) ).
% Inter_in_chain
thf(fact_1259_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X: A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( inv_on @ A @ B @ F2 @ A3 @ ( F2 @ X ) )
= X ) ) ) ).
% inv_on_f_f
thf(fact_1260_card__inverse,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ B @ A )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R4 ) )
= ( finite_card @ ( product_prod @ B @ A ) @ R4 ) ) ).
% card_inverse
thf(fact_1261_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_1262_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ( finite_card @ A @ S )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_1263_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F2 @ G @ X ) )
= ( F2 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_1264_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_1265_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_1266_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,X: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F2 @ X )
= ( product_apfst @ C @ A @ B @ G @ X ) )
= ( ( F2 @ ( product_fst @ C @ B @ X ) )
= ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_1267_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F2 @ X ) )
= ( F2 @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_1268_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F2 @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_1269_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R2 @ F2 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_1270_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F2 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_1271_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,X: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F2 @ X )
= ( product_apsnd @ C @ B @ A @ G @ X ) )
= ( ( F2 @ ( product_snd @ A @ C @ X ) )
= ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_1272_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F2 @ X ) )
= ( F2 @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_1273_converse__inv__image,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( converse @ A @ A @ ( inv_image @ B @ A @ R4 @ F2 ) )
= ( inv_image @ B @ A @ ( converse @ B @ B @ R4 ) @ F2 ) ) ).
% converse_inv_image
thf(fact_1274_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 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A3 )
& ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_1275_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_1276_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_1277_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_1278_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_insert
thf(fact_1279_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_1280_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_1281_card__Diff__insert,axiom,
! [A: $tType,A4: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ~ ( member @ A @ A4 @ B5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ B5 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_1282_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_1283_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_1284_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_1285_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_1286_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_1287_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_1288_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: A > C,G: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F2 @ G ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_1289_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_1290_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_1291_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F2: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apsnd @ D @ B @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( F2 @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_1292_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F2 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_1293_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,P3: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ P3 ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F2 @ P3 ) ) ) ).
% apsnd_apfst_commute
thf(fact_1294_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_1295_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_1296_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_1297_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A4: A,P3: product_prod @ A @ B] :
( ( A4
= ( product_fst @ A @ B @ P3 ) )
= ( ? [B6: B] :
( P3
= ( product_Pair @ A @ B @ A4 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_1298_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_1299_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y4: product_prod @ A @ B,Z5: product_prod @ A @ B] : Y4 = Z5 )
= ( ^ [S7: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S7 )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S7 )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_1300_prod__eqI,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Q4: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P3 )
= ( product_fst @ A @ B @ Q4 ) )
=> ( ( ( product_snd @ A @ B @ P3 )
= ( product_snd @ A @ B @ Q4 ) )
=> ( P3 = Q4 ) ) ) ).
% prod_eqI
thf(fact_1301_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_1302_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A8: A,X7: B] : ( P @ A8 @ X7 ) )
= ( ! [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% All_prod_contract
thf(fact_1303_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A8: A,X7: B] : ( P @ A8 @ X7 ) )
= ( ? [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% Ex_prod_contract
thf(fact_1304_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F2 @ ( product_apfst @ D @ C @ B @ G @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F2 @ G ) @ X ) ) ).
% apfst_compose
thf(fact_1305_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: C > B,G: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apsnd @ D @ C @ A @ G @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F2 @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_1306_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,A3: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( refl_on @ B @ ( A3 @ X3 ) @ ( R2 @ X3 ) ) )
=> ( refl_on @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ S ) ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% refl_on_INTER
thf(fact_1307_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( ( finite_card @ A @ A5 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_1308_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A4: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ A4 @ X3 ) )
=> ( ! [Y2: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X4 )
=> ( ord_less_eq @ A @ Y2 @ X6 ) )
=> ( ord_less_eq @ A @ Y2 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= A4 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1309_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,Z2: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ).
% cInf_greatest
thf(fact_1310_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_1311_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Z2: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X4 ) @ Z2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( ord_less @ A @ X3 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_1312_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: set @ B,F2: B > A,X: A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_1313_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_1314_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_1315_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_finite_insert
thf(fact_1316_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_1317_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_1318_surjective__pairing,axiom,
! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
( T5
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T5 ) @ ( product_snd @ A @ B @ T5 ) ) ) ).
% surjective_pairing
thf(fact_1319_card__insert__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert2 @ A @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_1320_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_1321_trans__inv__image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),F2: B > A] :
( ( trans @ A @ R2 )
=> ( trans @ B @ ( inv_image @ A @ B @ R2 @ F2 ) ) ) ).
% trans_inv_image
thf(fact_1322_Inter__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B5: set @ A] :
( ! [X8: set @ A] :
( ( member @ ( set @ A ) @ X8 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ B5 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B5 ) ) ) ).
% Inter_subset
thf(fact_1323_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_1324_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R4 )
= ( domain @ A @ B @ R4 ) ) ).
% fst_eq_Domain
thf(fact_1325_Domain__fst,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% Domain_fst
thf(fact_1326_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ A ),F2: A > B] :
( ( S
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A11: set @ A] :
( ( member @ ( set @ A ) @ A11 @ S )
=> ( inj_on @ A @ B @ F2 @ A11 ) )
=> ( inj_on @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_1327_trans__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( trans @ B @ ( R2 @ X3 ) ) )
=> ( trans @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% trans_INTER
thf(fact_1328_card__partition,axiom,
! [A: $tType,C6: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C6 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) )
=> ( ! [C3: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C6 )
=> ( ( finite_card @ A @ C3 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C6 )
=> ( ( member @ ( set @ A ) @ C22 @ C6 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C6 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1329_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_1330_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F2 @ A3 @ Y ) @ A3 ) ) ).
% inv_on_f_range
thf(fact_1331_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( F2 @ ( inv_on @ B @ A @ F2 @ A3 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_1332_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M: A,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_1333_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: set @ B,F2: B > A,C2: A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I )
=> ( ( F2 @ X2 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1334_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,A4: A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A4 @ ( complete_Inf_Inf @ A @ X4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X4 )
=> ( ord_less @ A @ A4 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_1335_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_1336_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( member @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_1337_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S )
=> ( S
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_1338_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_1339_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_1340_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_1341_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_1342_fst__image__mp,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B5: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) @ B5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A3 )
=> ( member @ A @ X @ B5 ) ) ) ).
% fst_image_mp
thf(fact_1343_fst__in__Field,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_fst @ A @ A ) @ R4 ) @ ( field2 @ A @ R4 ) ) ).
% fst_in_Field
thf(fact_1344_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_1345_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= ( lattic7752659483105999362nf_fin @ A @ X4 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_1346_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( insert2 @ A @ ( product_fst @ A @ B @ P5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_1347_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P5 ) @ ( product_fst @ A @ B @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_1348_equiv__proj,axiom,
! [A: $tType,A3: set @ A,R4: set @ ( product_prod @ A @ A ),Z2: product_prod @ A @ A] :
( ( equiv_equiv @ A @ A3 @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ Z2 @ R4 )
=> ( ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R4 ) @ ( product_fst @ A @ A ) @ Z2 )
= ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R4 ) @ ( product_snd @ A @ A ) @ Z2 ) ) ) ) ).
% equiv_proj
thf(fact_1349_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_1350_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_1351_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_1352_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P3: A,Q2: B > $o,Q4: B] :
( ( P @ P3 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q4 ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1353_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F2: C > A,P3: product_prod @ C @ B] :
( ( Q4
= ( product_apfst @ C @ A @ B @ F2 @ P3 ) )
=> ~ ! [X3: C,Y2: B] :
( ( P3
= ( product_Pair @ C @ B @ X3 @ Y2 ) )
=> ( Q4
!= ( product_Pair @ A @ B @ ( F2 @ X3 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_1354_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: A > B > $o,P3: product_prod @ B @ A] :
( ( P @ ( product_snd @ B @ A @ P3 ) @ ( product_fst @ B @ A @ P3 ) )
=> ~ ! [X3: B,Y2: A] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_1355_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P: B > C > $o,Q2: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_1356_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > C > $o,Q2: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_1357_card__Un__disjoint,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1358_card__Suc__Diff1,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1359_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add_left_cancel
thf(fact_1360_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ).
% add_right_cancel
thf(fact_1361_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_1362_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_1363_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_1364_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_1365_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_1366_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_1367_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_1368_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_1369_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_1370_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_1371_card__insert__disjoint,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1372_Suc__diff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ M ) )
= ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_1373_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus @ A @ I2 @ K )
= ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1374_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_1375_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B3: A,A4: A] :
( ( B5
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A4 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_1376_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_1377_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add.left_cancel
thf(fact_1378_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ).
% add.right_cancel
thf(fact_1379_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A8: A,B6: A] : ( plus_plus @ A @ B6 @ A8 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_1380_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_1381_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C2 ) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_1382_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C2 @ A4 ) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_1383_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1384_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1385_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1386_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_1387_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_1388_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ! [C3: A] :
( B3
!= ( plus_plus @ A @ A4 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_1389_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_1390_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B6: A] :
? [C5: A] :
( B6
= ( plus_plus @ A @ A8 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_1391_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_1392_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_1393_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J2 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1394_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1395_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1396_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1397_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_1398_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_1399_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_1400_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_1401_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,E4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ E4 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_1402_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_1403_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_1404_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1405_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1406_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1407_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_1408_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= C2 )
= ( A4
= ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_1409_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( A4
= ( minus_minus @ A @ C2 @ B3 ) )
= ( ( plus_plus @ A @ A4 @ B3 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_1410_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_1411_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_1412_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_1413_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1414_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ( plus_plus @ A @ C2 @ B3 )
= A4 )
=> ( C2
= ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_1415_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_1416_mlex__bound,axiom,
! [A4: nat,A3: nat,B3: nat,N: nat] :
( ( ord_less @ nat @ A4 @ A3 )
=> ( ( ord_less @ nat @ B3 @ N )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N ) @ B3 ) @ ( times_times @ nat @ A3 @ N ) ) ) ) ).
% mlex_bound
thf(fact_1417_mlex__fst__decrI,axiom,
! [A4: nat,A7: nat,B3: nat,N: nat,B4: nat] :
( ( ord_less @ nat @ A4 @ A7 )
=> ( ( ord_less @ nat @ B3 @ N )
=> ( ( ord_less @ nat @ B4 @ N )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A7 @ N ) @ B4 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_1418_mlex__snd__decrI,axiom,
! [A4: nat,A7: nat,B3: nat,B4: nat,N: nat] :
( ( A4 = A7 )
=> ( ( ord_less @ nat @ B3 @ B4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A7 @ N ) @ B4 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_1419_mlex__leI,axiom,
! [A4: nat,A7: nat,B3: nat,B4: nat,N: nat] :
( ( ord_less_eq @ nat @ A4 @ A7 )
=> ( ( ord_less_eq @ nat @ B3 @ B4 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A7 @ N ) @ B4 ) ) ) ) ).
% mlex_leI
thf(fact_1420_Inf__nat__def1,axiom,
! [K2: set @ nat] :
( ( K2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K2 ) @ K2 ) ) ).
% Inf_nat_def1
thf(fact_1421_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_1422_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_1423_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J2 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1424_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1425_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1426_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1427_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A4 )
= C2 )
= ( B3
= ( plus_plus @ A @ C2 @ A4 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1428_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_1429_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1430_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A4 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1431_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1432_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B3 ) @ A4 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1433_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B3 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1434_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1435_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% le_add_diff
thf(fact_1436_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_1437_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1438_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_1439_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_1440_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_1441_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( ord_less @ A @ A4 @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_1442_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ A4 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1443_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_1444_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ D3 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E4 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1445_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E4 ) @ C2 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1446_Suc__n__minus__m__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ M ) )
= ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1447_nat__in__between__eq_I1_J,axiom,
! [A4: nat,B3: nat] :
( ( ( ord_less @ nat @ A4 @ B3 )
& ( ord_less_eq @ nat @ B3 @ ( suc @ A4 ) ) )
= ( B3
= ( suc @ A4 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_1448_nat__in__between__eq_I2_J,axiom,
! [A4: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A4 @ B3 )
& ( ord_less @ nat @ B3 @ ( suc @ A4 ) ) )
= ( B3 = A4 ) ) ).
% nat_in_between_eq(2)
thf(fact_1449_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),F2: nat > A] :
( ( wf @ A @ R2 )
=> ~ ! [K3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ K3 ) ) @ ( F2 @ K3 ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_1450_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_1451_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S6: set @ ( A > $o ),X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1452_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E4 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1453_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E4 ) @ C2 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1454_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E4 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1455_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E4 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E4 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1456_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_1457_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B6: A,B8: set @ A] :
( ( A3
= ( insert2 @ A @ B6 @ B8 ) )
& ~ ( member @ A @ B6 @ B8 )
& ( ( finite_card @ A @ B8 )
= K )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1458_card__insert__if,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A3 ) )
= ( finite_card @ A @ A3 ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1459_card__le__Suc__iff,axiom,
! [A: $tType,N2: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ A3 ) )
= ( ? [A8: A,B8: set @ A] :
( ( A3
= ( insert2 @ A @ A8 @ B8 ) )
& ~ ( member @ A @ A8 @ B8 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ B8 ) )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1460_card_Oremove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1461_card_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1462_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ M @ ( suc @ N2 ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1463_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1464_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1465_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1466_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1467_nat__diff__add__eq2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1468_nat__diff__add__eq1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1469_nat__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1470_nat__mult__max__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_1471_nat__mult__max__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_1472_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1473_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1474_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1475_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1476_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1477_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1478_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N2 ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1479_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1480_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1481_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1482_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1483_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1484_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
= N2 ) ).
% nat_mult_1_right
thf(fact_1485_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1486_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1487_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N2 )
= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_1488_nat__eq__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1489_nat__eq__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1490_nat__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1491_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A8 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1492_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P2: A > B > $o,Q: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_1493_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P2: C > A > $o,Q: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P2 @ Q @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_1494_card__insert__le__m1,axiom,
! [A: $tType,N2: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1495_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( A4 != B3 )
& ( C2 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A4 @ D3 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1496_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y: A,X: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z2 ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_1497_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_1498_card__insert__disjoint_H,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ A3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_1499_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_1500_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_1501_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% add.right_neutral
thf(fact_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% add_0
thf(fact_1510_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_right
thf(fact_1511_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_left
thf(fact_1512_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_1513_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_1514_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_1515_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1516_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_0_right
thf(fact_1517_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_1518_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_zero
thf(fact_1519_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_1520_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_1521_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_1522_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_1523_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_1524_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_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_1532_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_1533_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_1541_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_1542_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_1543_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A4: A,C2: A] :
( ( ( times_times @ A @ A4 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1544_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B3: A] :
( ( C2
= ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1545_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A4: A] :
( ( ( times_times @ A @ C2 @ A4 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1546_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B3: A] :
( ( C2
= ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1547_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
= S ) ) ).
% image_add_0
thf(fact_1548_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_1549_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_1550_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_1551_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_1552_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1553_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1554_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1555_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1556_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1557_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_1558_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1559_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1560_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1561_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_1562_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_1563_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_1564_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_1565_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_1566_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ M @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_1567_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_1568_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_1569_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_1570_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_1571_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B3 @ C2 ) )
= ( A4 = B3 ) ) ) ) ).
% mult_right_cancel
thf(fact_1572_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% mult_left_cancel
thf(fact_1573_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_1574_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_1575_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_1576_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_1577_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [A8: A,B6: A] :
( ( minus_minus @ A @ A8 @ B6 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1578_mult__0,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1579_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1580_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A4: A,B3: A,C2: A,D3: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A4 = B3 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A4 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R2 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1581_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_1582_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% add_decreasing
thf(fact_1583_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_1584_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% add_decreasing2
thf(fact_1585_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_1586_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_1587_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_1588_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_1589_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_1590_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1591_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_1592_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_1593_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_1594_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_1595_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_1596_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_1597_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_1598_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1599_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1600_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1601_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_1602_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1603_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_1604_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_1605_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1606_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1607_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_1608_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_1609_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_1610_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B6: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A8 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1611_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1612_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ! [C3: A] :
( ( B3
= ( plus_plus @ A @ A4 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1626_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1627_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1628_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1629_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1630_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1631_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1632_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1633_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1634_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_1635_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_1636_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_1637_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B6: A] : ( ord_less @ A @ ( minus_minus @ A @ A8 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1638_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_1639_nat__compl__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_compl_induct
thf(fact_1640_nat__compl__induct_H,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_compl_induct'
thf(fact_1641_Suc__to__right,axiom,
! [N2: nat,M: nat] :
( ( ( suc @ N2 )
= M )
=> ( N2
= ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_1642_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1643_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1644_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1645_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1646_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_1647_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times @ nat @ M @ N2 ) )
=> ( ( N2
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1648_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1649_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1650_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_1651_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_1652_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_1653_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_1654_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1655_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1656_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1657_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1658_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1659_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1660_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1661_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1662_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1663_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1664_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1665_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1666_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1667_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1668_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_1669_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ A4 ) ) ) ) ).
% mult_left_le
thf(fact_1670_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_1671_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_1672_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_1673_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_1674_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_1675_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_1676_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1677_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1678_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1679_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1680_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1681_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1682_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1683_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1684_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1685_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1686_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1687_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1688_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_1689_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_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_1690_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_1691_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
=> ? [B2: A,B7: set @ A] :
( ( A3
= ( insert2 @ A @ B2 @ B7 ) )
& ~ ( member @ A @ B2 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_1692_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B6: A,B8: set @ A] :
( ( A3
= ( insert2 @ A @ B6 @ B8 ) )
& ~ ( member @ A @ B6 @ B8 )
& ( ( finite_card @ A @ B8 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B8
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1693_nz__le__conv__less,axiom,
! [K: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M ) ) ) ).
% nz_le_conv_less
thf(fact_1694_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M4: nat,N4: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N4 @ ( times_times @ nat @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1695_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_1696_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_1697_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_1698_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G4: B > nat,P5: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P5 ) ) @ ( G4 @ ( product_snd @ A @ B @ P5 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_1699_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_1700_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_1701_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_1702_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_1703_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1704_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F2 @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F2 @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_1705_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique5940410009612947441es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_1706_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_1707_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_1708_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_1709_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N2 ) ) ) ).
% enumerate_Suc'
thf(fact_1710_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N2 ) )
= ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1711_dvd__partition,axiom,
! [A: $tType,C6: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C6 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X3 ) ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C6 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_1712_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_1713_sum__le__card__Max,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F2 @ A3 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_1714_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_1715_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N2 )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_1716_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_1717_bdd__below__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( condit1013018076250108175_below @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_below_insert
thf(fact_1718_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1719_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1720_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1721_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1722_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A4 ) @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1723_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ ( times_times @ A @ C2 @ A4 ) ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1724_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_1725_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ( power_power @ A @ A4 @ M )
= ( power_power @ A @ A4 @ N2 ) )
= ( M = N2 ) ) ) ) ).
% power_inject_exp
thf(fact_1726_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1727_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_1728_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_1729_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N2 ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1730_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_1731_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A4 @ N2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1732_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N2: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1733_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,N2: nat] :
( ( power_power @ A @ ( times_times @ A @ A4 @ B3 ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ A4 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) ) ) ) ).
% power_mult_distrib
thf(fact_1734_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ A4 @ N2 ) @ A4 )
= ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% power_commutes
thf(fact_1735_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_1736_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
=> ( dvd_dvd @ A @ B3 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1737_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1738_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_1739_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
=> ( dvd_dvd @ A @ A4 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1740_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1741_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C2 )
=> ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1742_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B6: A,A8: A] :
? [K5: A] :
( A8
= ( times_times @ A @ B6 @ K5 ) ) ) ) ) ).
% dvd_def
thf(fact_1743_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_1744_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ~ ! [K3: A] :
( A4
!= ( times_times @ A @ B3 @ K3 ) ) ) ) ).
% dvdE
thf(fact_1745_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_1746_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_1747_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A4 ) ) ).
% one_dvd
thf(fact_1748_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( power_power @ A @ A4 @ ( times_times @ nat @ M @ N2 ) )
= ( power_power @ A @ ( power_power @ A @ A4 @ M ) @ N2 ) ) ) ).
% power_mult
thf(fact_1749_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N2: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N2 ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1750_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% dvd_power
thf(fact_1751_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_1752_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% one_le_power
thf(fact_1753_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_1754_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N2: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ N2 ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1755_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ A4 @ ( suc @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ N2 ) @ A4 ) ) ) ).
% power_Suc2
thf(fact_1756_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ A4 @ ( suc @ N2 ) )
= ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% power_Suc
thf(fact_1757_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B3 @ A4 )
= ( times_times @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1758_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A4 @ B3 )
= ( times_times @ A @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1759_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1760_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1761_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1762_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1763_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_1764_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A4: A] :
( ( power_power @ A @ A4 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1765_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( power_power @ A @ A4 @ ( plus_plus @ nat @ M @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ M ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% power_add
thf(fact_1766_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_1767_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: 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 @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1768_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N2 ) @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1769_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1770_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_1771_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_1772_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ~ ( ( A4
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B3
!= ( times_times @ A @ A4 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1773_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ ( suc @ N2 ) ) ) ) ) ).
% power_gt1
thf(fact_1774_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N: nat,A4: A] :
( ( ord_less @ nat @ N2 @ N )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N2 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1775_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ A @ ( power_power @ A @ A4 @ M ) @ ( power_power @ A @ A4 @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1776_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N: nat,A4: A] :
( ( ord_less_eq @ nat @ N2 @ N )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A4 @ N2 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_increasing
thf(fact_1777_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T5: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X6: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X6 @ T5 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D4 ) ) @ T5 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1778_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T5: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X6: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X6 @ T5 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D4 ) ) @ T5 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1779_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ B5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A3 )
& ( ord_less_eq @ A @ X6 @ B2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_1780_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_1781_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A,Y: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X4 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X4 ) @ Y )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( ord_less @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_1782_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_1783_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1784_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: 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 @ N2 ) ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ).
% power_Suc_less
thf(fact_1785_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: 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 @ N2 ) ) @ A4 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1786_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1787_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N: nat,A4: A] :
( ( ord_less @ nat @ N2 @ N )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1788_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N: nat,A4: A] :
( ( ord_less_eq @ nat @ N2 @ N )
=> ( ( 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 ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1789_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A4 @ M ) @ ( power_power @ A @ A4 @ N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1790_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ A @ A4 @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ).
% self_le_power
thf(fact_1791_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ) ).
% one_less_power
thf(fact_1792_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ B,F2: C > A,A3: set @ C,G: B > A] :
( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F2 @ A3 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B5 )
=> ? [X6: C] :
( ( member @ C @ X6 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_1793_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_1794_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B5 ) @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1795_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A4: A] :
( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X4 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ X4 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_1796_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X4: set @ A,A4: A] :
( ( condit1013018076250108175_below @ A @ X4 )
=> ( ( ( X4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ X4 ) )
= A4 ) )
& ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A4 @ X4 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ X4 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_1797_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_1798_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_1799_dvd__mult__cancel1,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N2 ) @ M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_1800_dvd__mult__cancel2,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M ) @ M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_1801_dvd__minus__add,axiom,
! [Q4: nat,N2: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q4 @ N2 )
=> ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ Q4 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1802_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B5 )
=> ( ord_less_eq @ A @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1803_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1804_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_1805_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M4: nat] :
( if @ A
@ ( M4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1806_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N2: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( power_power @ A @ A4 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A4 )
= ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% power_minus_mult
thf(fact_1807_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,B5: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ B5 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B5 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_1808_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_1809_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_1810_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I2: C,A3: set @ C,F2: C > B] :
( ( member @ C @ I2 @ A3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert2 @ C @ I2 @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A3 )
=> ( ord_less_eq @ B @ ( F2 @ I2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_1811_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_1812_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A4: B,F2: B > A] :
( ( finite_finite2 @ 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_1813_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_1814_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_1815_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_1816_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_1817_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I )
=> ( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I ) ) ) ) ) ) ).
% sum_pos
thf(fact_1818_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C6 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C6 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C6 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_1819_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_1820_bezout__add__strong__nat,axiom,
! [A4: nat,B3: nat] :
( ( A4
!= ( zero_zero @ nat ) )
=> ? [D2: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D2 @ A4 )
& ( dvd_dvd @ nat @ D2 @ B3 )
& ( ( times_times @ nat @ A4 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1821_bezout1__nat,axiom,
! [A4: nat,B3: nat] :
? [D2: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D2 @ A4 )
& ( dvd_dvd @ nat @ D2 @ B3 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A4 @ X3 ) @ ( times_times @ nat @ B3 @ Y2 ) )
= D2 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B3 @ X3 ) @ ( times_times @ nat @ A4 @ Y2 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_1822_bezout__add__nat,axiom,
! [A4: nat,B3: nat] :
? [D2: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D2 @ A4 )
& ( dvd_dvd @ nat @ D2 @ B3 )
& ( ( ( times_times @ nat @ A4 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ D2 ) )
| ( ( times_times @ nat @ B3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A4 @ Y2 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1823_bezout__lemma__nat,axiom,
! [D3: nat,A4: nat,B3: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D3 @ A4 )
=> ( ( dvd_dvd @ nat @ D3 @ B3 )
=> ( ( ( ( times_times @ nat @ A4 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y ) @ D3 ) )
| ( ( times_times @ nat @ B3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A4 @ Y ) @ D3 ) ) )
=> ? [X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D3 @ A4 )
& ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A4 @ B3 ) )
& ( ( ( times_times @ nat @ A4 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A4 @ B3 ) @ Y2 ) @ D3 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A4 @ B3 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A4 @ Y2 ) @ D3 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1824_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P3: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ P3 @ ( times_times @ A @ A4 @ B3 ) )
=> ~ ! [X3: A,Y2: A] :
( ( P3
= ( times_times @ A @ X3 @ Y2 ) )
=> ( ( dvd_dvd @ A @ X3 @ A4 )
=> ~ ( dvd_dvd @ A @ Y2 @ B3 ) ) ) ) ) ).
% dvd_productE
thf(fact_1825_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
=> ? [B11: A,C9: A] :
( ( A4
= ( times_times @ A @ B11 @ C9 ) )
& ( dvd_dvd @ A @ B11 @ B3 )
& ( dvd_dvd @ A @ C9 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_1826_card__Min__le__sum,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F2 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ).
% card_Min_le_sum
thf(fact_1827_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_1828_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1829_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1830_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ).
% Min_in
thf(fact_1831_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1832_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1833_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1834_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
=> ! [A12: A] :
( ( member @ A @ A12 @ A3 )
=> ( ord_less_eq @ A @ X @ A12 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_1835_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1836_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_1837_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ A4 @ B2 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ A4 @ A3 ) )
= A4 ) ) ) ) ).
% Min_insert2
thf(fact_1838_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X4: set @ A] :
( ( finite_finite2 @ A @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X4 )
= ( lattic643756798350308766er_Min @ A @ X4 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_1839_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Min_Inf
thf(fact_1840_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: set @ A,N: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
=> ( ( M2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N ) @ ( lattic643756798350308766er_Min @ A @ M2 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_1841_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1842_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_1843_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_1844_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_1845_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_finite2 @ A @ A3 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_1846_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less @ A @ ( F2 @ I3 ) @ K2 ) )
=> ( ( 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 ) ) @ K2 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_1847_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,N2: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N2 @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_1848_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B5: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B2 ) ) )
=> ( ! [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 @ B5 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_1849_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_1850_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
= A4 ) ) ).
% add.inverse_inverse
thf(fact_1851_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1852_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1853_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_1854_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_1855_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_1856_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_1857_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_1858_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1859_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_1860_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_1861_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_1862_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_1863_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_1864_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_1865_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_1866_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_1867_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_1868_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_1869_inter__compl__diff__conv,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ).
% inter_compl_diff_conv
thf(fact_1870_Diff__Compl,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_Compl
thf(fact_1871_Compl__Diff__eq,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ B5 ) ) ).
% Compl_Diff_eq
thf(fact_1872_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_1873_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_1874_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_1875_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_1876_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_1877_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_1878_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_1879_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_1880_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_1881_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_1882_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_1883_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_1884_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_1885_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_1886_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_1887_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_1888_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_1889_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_1890_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_1891_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_1892_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_1893_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_1894_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_1895_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_1896_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_1897_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_mult
thf(fact_1898_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_1899_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_1900_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( semiring_1_of_nat @ A @ N2 )
= ( one_one @ A ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_1901_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_1902_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_1903_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_1904_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_1905_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_1906_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_1907_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_1908_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_1909_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_1910_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_1911_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat,A4: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ A4 ) )
= A4 ) ) ).
% left_minus_one_mult_self
thf(fact_1912_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% of_nat_Suc
thf(fact_1913_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_1914_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_1915_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_1916_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_1917_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_1918_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
!= ( one_one @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1919_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_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_vimage__Compl,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( vimage @ A @ B @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) ) ) ).
% vimage_Compl
thf(fact_1932_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_1933_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A8: assn,B6: assn] : ( inf_inf @ assn @ A8 @ ( uminus_uminus @ assn @ B6 ) ) ) ) ).
% minus_assn_def
thf(fact_1934_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_1935_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B5 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A16: set @ B] :
( ( member @ ( set @ B ) @ A16 @ B5 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B5 )
=> ( ( A16 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A16 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B5 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A8: A,B6: A] : ( plus_plus @ A @ A8 @ ( uminus_uminus @ A @ B6 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1947_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A8: A,B6: A] : ( plus_plus @ A @ A8 @ ( uminus_uminus @ A @ B6 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1948_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B5: A,K: A,B3: A,A4: A] :
( ( B5
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( minus_minus @ A @ A4 @ B5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1949_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_1950_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_1951_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_1952_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_1953_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_1954_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_1955_Compl__partition2,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_1956_Compl__partition,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_1957_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1958_Compl__Int,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% Compl_Int
thf(fact_1959_Compl__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% Compl_Un
thf(fact_1960_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K5: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K5 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K5 ) @ A8 ) @ ( one_one @ A ) ) @ K5 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1961_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: 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 @ N2 ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% gbinomial_index_swap
thf(fact_1962_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C6 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C6 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C6 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_1963_Gcd__remove0__nat,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( gcd_Gcd @ nat @ M2 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M2 @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_1964_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_1965_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_1966_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R4: A > A > $o,S: set @ B,H2: B > A,G: B > A] :
( ( R4 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X12: A,Y12: A,X23: A,Y23: A] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( times_times @ A @ X12 @ Y12 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R4 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_1967_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1968_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T6: set @ C,S: set @ B,I2: C > B,J2: B > C,T3: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( member @ C @ ( J2 @ A6 ) @ ( minus_minus @ ( set @ C ) @ T3 @ T6 ) ) )
=> ( ! [B2: C] :
( ( member @ C @ B2 @ ( minus_minus @ ( set @ C ) @ T3 @ T6 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: C] :
( ( member @ C @ B2 @ ( minus_minus @ ( set @ C ) @ T3 @ T6 ) )
=> ( member @ B @ ( I2 @ B2 ) @ ( minus_minus @ ( set @ B ) @ S @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B2: C] :
( ( member @ C @ B2 @ T6 )
=> ( ( H2 @ B2 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% power_minus
thf(fact_1976_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1977_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_1978_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ A3 ) @ B5 )
= ( ord_less_eq @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R2 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_1979_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I: set @ A,I2: A,F2: A > B] :
( ( finite_finite2 @ A @ I )
=> ( ( member @ A @ I2 @ I )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I2 ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_1980_rel__restrict__compl,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R4 @ A3 ) @ ( rel_restrict @ A @ R4 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_1981_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I )
=> ( ( I
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I ) ) ) ) ) ) ).
% less_1_prod
thf(fact_1982_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B,Y2: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( member @ B @ Y2 @ A3 )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_1983_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_1984_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B2: B] :
( ( member @ B @ B2 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H2 @ B2 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1985_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B2: B] :
( ( member @ B @ B2 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H2 @ B2 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1986_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_1987_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_1988_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1989_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1990_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_1991_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_1992_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( H2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T3 ) )
=> ( ( G @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S @ T3 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T3 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_1993_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G4 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_1994_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ K2 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K2 ) ) ) ) ).
% sum_bounded_above
thf(fact_1995_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K2: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ K2 @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_1996_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_1997_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_1998_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_1999_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_2000_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2001_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2002_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2003_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_2004_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_2005_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B5 @ A3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_2006_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_2007_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_2008_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A4: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A4 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A4 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2009_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2010_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2011_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_2012_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_2013_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_2014_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N2: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% power_minus'
thf(fact_2015_zmult__zless__mono2__lemma,axiom,
! [I2: int,J2: int,K: nat] :
( ( ord_less @ int @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I2 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_2016_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( A3 = B5 ) ) ).
% Compl_eq_Compl_iff
thf(fact_2017_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_2018_ComplI,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_2019_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( times_times @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_2020_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_2021_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% divide_divide_eq_right
thf(fact_2022_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_2023_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% div_by_1
thf(fact_2024_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_2025_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_2026_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_2027_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2028_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2029_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2030_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2031_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2032_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_2033_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_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_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_2058_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_2059_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_2060_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_2061_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_2062_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_2063_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_2064_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N2: int] :
( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_2065_zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_2066_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_2067_ComplD,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C2 @ A3 ) ) ).
% ComplD
thf(fact_2068_zdvd__mono,axiom,
! [K: int,M: int,T5: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T5 )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T5 ) ) ) ) ).
% zdvd_mono
thf(fact_2069_zdvd__mult__cancel,axiom,
! [K: int,M: int,N2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M @ N2 ) ) ) ).
% zdvd_mult_cancel
thf(fact_2070_zdvd__reduce,axiom,
! [K: int,N2: int,M: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N2 @ ( times_times @ int @ K @ M ) ) )
= ( dvd_dvd @ int @ K @ N2 ) ) ).
% zdvd_reduce
thf(fact_2071_zdvd__period,axiom,
! [A4: int,D3: int,X: int,T5: int,C2: int] :
( ( dvd_dvd @ int @ A4 @ D3 )
=> ( ( dvd_dvd @ int @ A4 @ ( plus_plus @ int @ X @ T5 ) )
= ( dvd_dvd @ int @ A4 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D3 ) ) @ T5 ) ) ) ) ).
% zdvd_period
thf(fact_2072_unique__quotient__lemma__neg,axiom,
! [B3: int,Q5: int,R7: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R7 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( ord_less @ int @ B3 @ R7 )
=> ( ord_less_eq @ int @ Q4 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_2073_unique__quotient__lemma,axiom,
! [B3: int,Q5: int,R7: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q5 ) @ R7 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R7 )
=> ( ( ord_less @ int @ R7 @ B3 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ord_less_eq @ int @ Q5 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_2074_zdiv__mono2__neg__lemma,axiom,
! [B3: int,Q4: int,R2: int,B4: int,Q5: int,R7: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R7 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R7 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B3 )
=> ( ord_less_eq @ int @ Q5 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_2075_zdiv__mono2__lemma,axiom,
! [B3: int,Q4: int,R2: int,B4: int,Q5: int,R7: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R7 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R7 ) )
=> ( ( ord_less @ int @ R7 @ B4 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B3 )
=> ( ord_less_eq @ int @ Q4 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_2076_int__div__pos__eq,axiom,
! [A4: int,B3: int,Q4: int,R2: int] :
( ( A4
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( divide_divide @ int @ A4 @ B3 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_2077_int__div__neg__eq,axiom,
! [A4: int,B3: int,Q4: int,R2: int] :
( ( A4
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( divide_divide @ int @ A4 @ B3 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_2078_q__pos__lemma,axiom,
! [B4: int,Q5: int,R7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R7 ) )
=> ( ( ord_less @ int @ R7 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_2079_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide @ int @ N2 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_2080_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus @ int @ X6 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_2081_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_2082_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K: int] :
( ( ord_less @ int @ I2 @ J2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I2 ) @ ( times_times @ int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_2083_minusinfinity,axiom,
! [D3: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D3 ) ) ) )
=> ( ? [Z7: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z7 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_2084_plusinfinity,axiom,
! [D3: int,P7: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K3: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D3 ) ) ) )
=> ( ? [Z7: int] :
! [X3: int] :
( ( ord_less @ int @ Z7 @ X3 )
=> ( ( P @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [X_12: int] : ( P7 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_2085_decr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus @ int @ X6 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_2086_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_2087_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_2088_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_2089_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_2090_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_2091_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_2092_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( divide_divide @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_2093_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z2: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_2094_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z2: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_2095_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A4: A,B3: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_2096_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C2: A,A4: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_2097_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A4: A,B3: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2098_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C2: A,A4: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2099_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W @ Z2 ) )
= ( ( times_times @ A @ X @ Z2 )
= ( times_times @ A @ W @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2100_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ( divide_divide @ A @ B3 @ C2 )
= A4 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_2101_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_2102_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B3
= ( times_times @ A @ A4 @ C2 ) )
=> ( ( divide_divide @ A @ B3 @ C2 )
= A4 ) ) ) ) ).
% divide_eq_imp
thf(fact_2103_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C2 )
= B3 )
=> ( A4
= ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_2104_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ C2 )
= A4 )
= ( B3
= ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2105_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A4
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( times_times @ A @ A4 @ C2 )
= B3 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2106_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_2107_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,D3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( dvd_dvd @ A @ D3 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( divide_divide @ A @ C2 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2108_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 )
=> ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2109_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ C2 ) @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2110_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2111_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_2112_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_2113_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2114_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2115_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= ( divide_divide @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_2116_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ N2 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N2 ) ) ) ) ).
% power_one_over
thf(fact_2117_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2118_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2119_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2120_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_2121_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2122_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2123_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_2124_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_2125_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_2126_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2127_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2128_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_2129_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_2130_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_2131_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_2132_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_2133_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_2134_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_2135_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_2136_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_2137_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_2138_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2139_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_2140_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_2141_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_2142_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_2143_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) ) )
= ( ( times_times @ A @ C2 @ B3 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2144_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) )
= C2 )
= ( ( uminus_uminus @ A @ A4 )
= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2145_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) )
= A4 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B3 )
= ( times_times @ A @ A4 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2146_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
= ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ C2 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2147_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_2148_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= ( divide_divide @ A @ D3 @ C2 ) )
= ( ( times_times @ A @ B3 @ C2 )
= ( times_times @ A @ A4 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2149_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2150_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2151_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= C2 )
= ( B3
= ( times_times @ A @ C2 @ A4 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2152_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_2153_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2154_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2155_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% unit_div_commute
thf(fact_2156_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2157_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( A4
= ( divide_divide @ A @ C2 @ B3 ) )
= ( ( times_times @ A @ A4 @ B3 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_2158_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= C2 )
= ( A4
= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2159_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_2160_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_2161_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E4 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E4 ) ) ) ).
% nat_approx_posE
thf(fact_2162_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2163_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2164_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_2165_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_2166_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2167_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2168_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_2169_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_2170_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_2171_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2172_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_2173_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_2174_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_2175_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_2176_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2177_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2178_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2179_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2180_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2181_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2182_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_2183_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_2184_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_2185_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_2186_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_2187_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ~ ( ( A4
!= ( zero_zero @ A ) )
=> ! [B2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A4 )
= B2 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B2 )
= A4 )
=> ( ( ( times_times @ A @ A4 @ B2 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A4 )
!= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2188_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_2189_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_2190_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( suc @ N2 ) )
= ( times_times @ A @ A4 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% pochhammer_rec
thf(fact_2191_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) ) ) ).
% pochhammer_rec'
thf(fact_2192_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( suc @ N2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A4 @ N2 ) @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2193_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N2: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N2 @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_2194_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R2: A,S2: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_2195_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2196_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2197_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2198_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2199_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2200_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2201_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N2: nat,Z2: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2202_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F2: B > A,A4: B] :
( ( finite_finite2 @ 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_2203_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F2: B > A,K2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( divide_divide @ A @ K2 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K2 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_2204_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A3: set @ B,B5: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
=> ( ( F2 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_2205_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_2206_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_2207_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2208_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_2209_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_2210_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C2 ) @ A4 ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self4
thf(fact_2211_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B3 ) @ A4 ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self3
thf(fact_2212_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self2
thf(fact_2213_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self1
thf(fact_2214_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_2215_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2216_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% div_mult_mult2
thf(fact_2217_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% div_mult_mult1
thf(fact_2218_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_2219_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_2220_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_2221_zdiv__zmult2__eq,axiom,
! [C2: int,A4: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A4 @ ( times_times @ int @ B3 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A4 @ B3 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_2222_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I2 @ N2 ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_2223_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_2224_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_2225_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( divide_divide @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% div_mult2_eq'
thf(fact_2226_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N2 )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2227_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_2228_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_2229_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N2 @ Q4 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_2230_div__nat__eqI,axiom,
! [N2: nat,Q4: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q4 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q4 ) ) )
=> ( ( divide_divide @ nat @ M @ N2 )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_2231_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_2232_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_2233_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_2234_split__div_H,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q6: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q6 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_2235_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,N2: nat,M: nat] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A4 @ M ) @ ( power_power @ A @ A4 @ N2 ) )
= ( power_power @ A @ A4 @ ( minus_minus @ nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A4 @ M ) @ ( power_power @ A @ A4 @ N2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_2236_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_2237_int__ops_I7_J,axiom,
! [A4: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A4 @ B3 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% int_ops(7)
thf(fact_2238_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K5 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A8 ) @ K5 ) ) @ ( semiring_char_0_fact @ A @ K5 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_2239_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K5: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A8 @ ( semiring_1_of_nat @ A @ K5 ) ) @ ( one_one @ A ) ) @ K5 ) @ ( semiring_char_0_fact @ A @ K5 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_2240_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q4
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_2241_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% pochhammer_same
thf(fact_2242_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2243_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_2244_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_2245_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_Suc
thf(fact_2246_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_1
thf(fact_2247_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_2248_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_2249_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2250_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% choose_dvd
thf(fact_2251_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M4: nat] :
( if @ A
@ ( M4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M4 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2252_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2253_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2254_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2255_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_2256_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_2257_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2258_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2259_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M: num,N2: num] :
( ( power_power @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N2 ) )
= ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% power_mult_numeral
thf(fact_2260_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_2261_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ).
% numeral_times_numeral
thf(fact_2262_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V: num,W: num,Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Z2 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_2263_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M: num,N2: num,B3: A] :
( ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ N2 ) ) @ B3 ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N2 ) ) ) @ B3 ) ) ) ).
% power_add_numeral2
thf(fact_2264_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M: num,N2: num] :
( ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ N2 ) ) )
= ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).
% power_add_numeral
thf(fact_2265_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2266_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_2267_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_2268_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_2269_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_2270_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2271_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_2272_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_2273_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B3: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_2274_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_2275_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B3: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_2276_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_2277_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_2278_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_2279_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_2280_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_2281_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self1
thf(fact_2291_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self2
thf(fact_2292_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B3 ) @ A4 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self3
thf(fact_2293_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C2 ) @ A4 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self4
thf(fact_2294_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_2295_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_2296_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_2297_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) @ A4 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2298_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W: num] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2299_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W: num,A4: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) )
= A4 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2300_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,W: num] :
( ( A4
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2301_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W: num] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2302_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) @ A4 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2303_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_2304_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_2305_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W: num] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2306_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2307_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W: num,A4: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A4 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_2308_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,W: num] :
( ( A4
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_2309_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2310_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W: num] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2311_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_2312_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_2313_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_2314_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2315_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_2316_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_2317_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A4 @ C2 ) @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_2318_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% mult_mod_right
thf(fact_2319_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_2320_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,A7: A,B3: A,B4: A] :
( ( ( modulo_modulo @ A @ A4 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A7 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_2321_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A4 @ C2 ) @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_2322_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A4 ) @ C2 )
=> ( ( 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 @ C2 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2331_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_2332_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_2333_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_2334_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_2335_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ( modulo_modulo @ A @ A4 @ C2 )
= ( modulo_modulo @ A @ B3 @ C2 ) )
=> ~ ! [D2: A] :
( B3
!= ( plus_plus @ A @ A4 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_2336_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% one_le_numeral
thf(fact_2337_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_2338_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_2339_zmod__eq__0__iff,axiom,
! [M: int,D3: int] :
( ( ( modulo_modulo @ int @ M @ D3 )
= ( zero_zero @ int ) )
= ( ? [Q6: int] :
( M
= ( times_times @ int @ D3 @ Q6 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_2340_zmod__eq__0D,axiom,
! [M: int,D3: int] :
( ( ( modulo_modulo @ int @ M @ D3 )
= ( zero_zero @ int ) )
=> ? [Q7: int] :
( M
= ( times_times @ int @ D3 @ Q7 ) ) ) ).
% zmod_eq_0D
thf(fact_2341_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2342_abs__zmult__eq__1,axiom,
! [M: int,N2: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N2 ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_2343_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ N2 )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_2344_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N2: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_2345_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_2346_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_2347_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_2348_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_2349_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2350_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_2351_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_2352_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_2353_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_2354_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_2355_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_2356_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_2357_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_2358_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_2359_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_2360_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_2361_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ C2 )
= ( plus_plus @ A @ A4 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2362_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ C2 )
= ( plus_plus @ A @ A4 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2363_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_2364_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_2365_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_2366_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_2367_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_2368_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_le_numeral
thf(fact_2369_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_2370_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_2371_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_2372_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2373_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2374_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_2375_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_2376_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_2377_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_less_numeral
thf(fact_2378_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_2379_div__mod__decomp__int,axiom,
! [A3: int,N2: int] :
( A3
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ int @ A3 @ N2 ) ) ) ).
% div_mod_decomp_int
thf(fact_2380_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_2381_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E4 ) ) ) ) ).
% cSup_asclose
thf(fact_2382_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E4 ) ) ) ) ).
% cInf_asclose
thf(fact_2383_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2384_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2385_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2386_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2387_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2388_zdvd__mult__cancel1,axiom,
! [M: int,N2: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N2 ) @ M )
= ( ( abs_abs @ int @ N2 )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_2389_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2390_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2391_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2392_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2393_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2394_split__zmod,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2395_int__mod__neg__eq,axiom,
! [A4: int,B3: int,Q4: int,R2: int] :
( ( A4
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( modulo_modulo @ int @ A4 @ B3 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2396_int__mod__pos__eq,axiom,
! [A4: int,B3: int,Q4: int,R2: int] :
( ( A4
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( modulo_modulo @ int @ A4 @ B3 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2397_zmod__zmult2__eq,axiom,
! [C2: int,A4: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A4 @ ( times_times @ int @ B3 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A4 @ B3 ) @ C2 ) ) @ ( modulo_modulo @ int @ A4 @ B3 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_2398_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_2399_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_2400_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_2401_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_2402_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A15: int,A24: int,A32: product_prod @ int @ int] :
( ? [K5: int] :
( ( A15 = K5 )
& ( A24
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K5 ) ) )
| ? [L2: int,K5: int,Q6: int] :
( ( A15 = K5 )
& ( A24 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K5
= ( times_times @ int @ Q6 @ L2 ) ) )
| ? [R5: int,L2: int,K5: int,Q6: int] :
( ( A15 = K5 )
& ( A24 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) )
& ( K5
= ( plus_plus @ int @ ( times_times @ int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_2403_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_2404_Suc__mod__mult__self4,axiom,
! [N2: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ M ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self4
thf(fact_2405_Suc__mod__mult__self3,axiom,
! [K: nat,N2: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self3
thf(fact_2406_Suc__mod__mult__self2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N2 @ K ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self2
thf(fact_2407_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N2 ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self1
thf(fact_2408_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_2409_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_2410_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_2411_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_2412_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_2413_Suc__times__numeral__mod__eq,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N2 ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_2414_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_2415_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_2416_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_2417_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_2418_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_2419_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_2420_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_2421_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_2422_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_2423_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_2424_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_2425_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_2426_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_2427_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_2428_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_2429_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_2430_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_2431_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_2432_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_2433_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_2434_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_2435_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L3: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_sgnE
thf(fact_2436_mod__eq__0D,axiom,
! [M: nat,D3: nat] :
( ( ( modulo_modulo @ nat @ M @ D3 )
= ( zero_zero @ nat ) )
=> ? [Q7: nat] :
( M
= ( times_times @ nat @ D3 @ Q7 ) ) ) ).
% mod_eq_0D
thf(fact_2437_nat__mod__eq__iff,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N2 @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_2438_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_2439_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K5: A] : ( times_times @ A @ K5 @ ( sgn_sgn @ A @ K5 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_2440_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_2441_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_2442_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_2443_nat__mod__eq__lemma,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q7: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q7 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2444_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ~ ! [S8: nat] :
( N2
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S8 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2445_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ~ ! [S8: nat] :
( M
!= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q4 @ S8 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2446_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_2447_div__mod__decomp,axiom,
! [A3: nat,N2: nat] :
( A3
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ nat @ A3 @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_2448_mod__mult2__eq,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ).
% mod_mult2_eq
thf(fact_2449_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M4: nat,N4: nat] : ( minus_minus @ nat @ M4 @ ( times_times @ nat @ ( divide_divide @ nat @ M4 @ N4 ) @ N4 ) ) ) ) ).
% modulo_nat_def
thf(fact_2450_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_2451_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_2452_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_2453_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_2454_split__mod,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2455_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_2456_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_2457_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_2458_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_2459_Suc__times__mod__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N2 ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_2460_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_2461_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q7: int] :
( ( A33
= ( product_Pair @ int @ int @ Q7 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q7 @ A22 ) ) ) )
=> ~ ! [R: int,Q7: int] :
( ( A33
= ( product_Pair @ int @ int @ Q7 @ R ) )
=> ( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q7 @ A22 ) @ R ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_2462_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_2463_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y3: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y3
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X2 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_2464_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_2465_divide__int__unfold,axiom,
! [L: int,K: int,N2: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N2 ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N2 )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_2466_modulo__int__unfold,axiom,
! [L: int,K: int,N2: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N2
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_2467_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_2468_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_2469_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_2470_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_2471_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_2472_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_2473_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q2: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q2 ) ) ) ) ).
% of_bool_conj
thf(fact_2474_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_2475_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_2476_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ( P3
=> ( P @ ( one_one @ A ) ) )
& ( ~ P3
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_2477_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P3
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_2485_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_2486_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: 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 @ P3 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P3 ) ) ) ).
% ceiling_divide_lower
thf(fact_2487_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P3 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P3 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_2488_semiring__norm_I12_J,axiom,
! [N2: num] :
( ( times_times @ num @ one2 @ N2 )
= N2 ) ).
% semiring_norm(12)
thf(fact_2489_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_2490_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: num] :
( ( ( numeral_numeral @ A @ N2 )
= ( one_one @ A ) )
= ( N2 = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_2491_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N2 ) )
= ( one2 = N2 ) ) ) ).
% one_eq_numeral_iff
thf(fact_2492_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_2493_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_2494_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z2: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z2 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_mult
thf(fact_2495_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N2 @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_2496_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ one2 @ N2 ) ) ) ).
% one_less_numeral_iff
thf(fact_2497_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N2: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N2 = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_2498_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N2: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( N2 = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_2499_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
= ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_2500_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_2501_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_2502_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_2503_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N2 @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_2504_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ).
% one_plus_numeral
thf(fact_2505_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_2506_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_2507_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_2508_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_2509_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_2510_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_2511_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_2512_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2520_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2521_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_2522_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_2523_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_2524_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_2525_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T5: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T5 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T5 )
& ( ord_less @ A @ T5 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_2526_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_2527_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_2528_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_2529_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_2530_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_2531_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T5: A] :
( ( P @ ( archimedean_ceiling @ A @ T5 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T5 )
& ( ord_less_eq @ A @ T5 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_2532_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_2533_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_2534_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_2535_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_2536_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: 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 @ P3 @ Q4 ) ) ) @ Q4 ) @ P3 ) ) ) ).
% floor_divide_lower
thf(fact_2537_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P3 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P3 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2538_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_2539_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2540_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2541_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q4: int,R2: 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 @ R2 ) )
=> ( 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 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_2542_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_double
thf(fact_2543_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( 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 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ).
% pochhammer_double
thf(fact_2544_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ord_less @ nat @ N2 @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2545_semiring__norm_I13_J,axiom,
! [M: num,N2: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N2 ) ) ) ) ).
% semiring_norm(13)
thf(fact_2546_num__double,axiom,
! [N2: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_2547_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_2548_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_2549_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_2550_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_2551_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2552_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_2553_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_2554_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_2555_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_2556_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_2557_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_2558_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_2559_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_2560_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_2561_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_2562_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_2563_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_2564_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( power_power @ A @ A4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_2565_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_2566_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_2567_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_2568_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_2569_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_2570_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_2571_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_2572_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2573_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_2574_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_2575_odd__two__times__div__two__nat,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_2576_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_2577_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_2578_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_2579_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_2580_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ~ ! [B2: A] :
( A4
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% evenE
thf(fact_2581_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_2582_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_2583_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_2584_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_2585_Suc__double__not__eq__double,axiom,
! [M: nat,N2: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% Suc_double_not_eq_double
thf(fact_2586_double__not__eq__Suc__double,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_2587_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ A4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( power_power @ A @ ( power_power @ A @ A4 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_2588_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_2589_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_2590_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_2591_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_2592_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_2593_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_2594_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ~ ! [B2: A] :
( A4
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2595_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_2596_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_2597_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_2598_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_2599_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% zero_le_even_power'
thf(fact_2600_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_2601_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_2602_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,B2: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( 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 @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% bits_induct
thf(fact_2603_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A4: A,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N2 ) )
= ( power_power @ A @ A4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% minus_power_mult_self
thf(fact_2604_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N2: nat] :
( ( power_power @ A @ A4 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( times_times @ A @ A4 @ ( power_power @ A @ ( power_power @ A @ A4 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_2605_nat__bit__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_2606_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_2607_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_2608_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_2609_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A4 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2610_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N2: nat] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2611_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat,A4: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2612_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_2613_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N2: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_2614_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2615_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2616_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_2617_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% even_mask_div_iff'
thf(fact_2618_pos__zdiv__mult__2,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A4 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) )
= ( divide_divide @ int @ B3 @ A4 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2619_neg__zdiv__mult__2,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq @ int @ A4 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A4 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2620_pos__zmod__mult__2,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A4 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ A4 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2621_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% even_mask_div_iff
thf(fact_2622_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N2 @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_2623_neg__zmod__mult__2,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq @ int @ A4 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A4 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2624_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( 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 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_2625_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_2626_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q4: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_2627_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_2628_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y3: A] :
( ( S
= ( insert2 @ A @ X2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y3 ) ) ) ) ).
% card_2_iff
thf(fact_2629_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_2630_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_2631_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2632_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2633_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2634_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A8: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2635_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_2636_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_2637_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_2638_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N2: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N2 ) ) ) ).
% round_unique'
thf(fact_2639_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_2640_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_2641_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_2642_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_2643_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2644_signed__take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_2645_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_2646_signed__take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_2647_signed__take__bit__mult,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( times_times @ int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_2648_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_2649_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2650_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_2651_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_2652_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_2653_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_2654_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K5: int,L2: int] :
( if @ int
@ ( ( member @ int @ K5 @ ( 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 ) ) @ K5 )
& ~ ( 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 ) ) @ K5 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_2655_signed__take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2656_concat__bit__Suc,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_2657_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2658_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A8: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2659_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_2660_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_2661_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_2662_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_2663_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_2664_semiring__norm_I14_J,axiom,
! [M: num,N2: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N2 ) ) ) ) ).
% semiring_norm(14)
thf(fact_2665_semiring__norm_I15_J,axiom,
! [M: num,N2: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N2 ) ) ) ).
% semiring_norm(15)
thf(fact_2666_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_2667_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_2668_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_2669_semiring__norm_I16_J,axiom,
! [M: num,N2: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N2 ) @ ( bit0 @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_2670_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_2671_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_2672_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_2673_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% take_bit_of_1
thf(fact_2674_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_2675_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_2676_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_2677_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_2678_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_2679_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_of_exp
thf(fact_2680_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2681_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2682_signed__take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_2683_take__bit__mult,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( times_times @ int @ K @ L ) ) ) ).
% take_bit_mult
thf(fact_2684_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_2685_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_2686_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2687_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_2688_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2689_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_2690_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_2691_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y3: A,Z3: A] :
( ( S
= ( insert2 @ A @ X2 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y3 )
& ( Y3 != Z3 )
& ( X2 != Z3 ) ) ) ) ).
% card_3_iff
thf(fact_2692_take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_2693_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K5: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K5 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_2694_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_2695_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A4 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( 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_2696_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_2697_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,N2: nat] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A4 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A4 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2698_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K5: int,L2: int] :
( if @ int
@ ( ( K5
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K5
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K5
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_2699_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2700_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( ( ord_less @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2701_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( ( ord_less_eq @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_2702_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2703_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_2704_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_2705_numeral__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K: num,L: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).
% numeral_mod_numeral
thf(fact_2706_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num] :
( ( unique8689654367752047608divmod @ A @ M @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2707_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2708_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2709_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N2 ) ) ) ) ).
% one_div_numeral
thf(fact_2710_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N2 ) ) ) ) ).
% one_mod_numeral
thf(fact_2711_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_2712_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2713_snd__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% snd_divmod
thf(fact_2714_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_2715_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M4: num,N4: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M4 ) @ ( numeral_numeral @ A @ N4 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M4 ) @ ( numeral_numeral @ A @ N4 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2716_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_2717_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_2718_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M4: nat,N4: nat] :
( if @ nat
@ ( ( M4
= ( zero_zero @ nat ) )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2719_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M4: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2720_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 ) )
=> ( ! [K3: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K3 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K3 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_2721_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M4: num,N4: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M4 @ N4 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M4 ) ) @ ( unique1321980374590559556d_step @ A @ N4 @ ( unique8689654367752047608divmod @ A @ M4 @ ( bit0 @ N4 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2722_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_2723_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_2724_take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2725_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K5: nat] : ( if @ nat @ ( ord_less @ nat @ N4 @ K5 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K5 ) ) @ ( binomial @ N4 @ ( minus_minus @ nat @ N4 @ K5 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N4 @ K5 ) @ ( one_one @ nat ) ) @ N4 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K5 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2726_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K5: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K5
@ ( if @ int
@ ( ( sgn_sgn @ int @ K5 )
= ( sgn_sgn @ int @ L2 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K5 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L2 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K5 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K5 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_2727_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ).
% mask_numeral
thf(fact_2728_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K5: int,L2: int] :
( if @ int
@ ( ( K5
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K5
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K5
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_2729_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_2730_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_2731_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_2732_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_2733_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_2734_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_2735_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_2736_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_2737_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_2738_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2739_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2740_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2741_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_2742_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_2743_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_2744_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_2745_Suc__times__binomial__eq,axiom,
! [N2: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_2746_Suc__times__binomial,axiom,
! [K: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% Suc_times_binomial
thf(fact_2747_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_2748_Suc__times__binomial__add,axiom,
! [A4: nat,B3: nat] :
( ( times_times @ nat @ ( suc @ A4 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A4 @ B3 ) ) @ ( suc @ A4 ) ) )
= ( times_times @ nat @ ( suc @ B3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A4 @ B3 ) ) @ A4 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2749_binomial__fact__lemma,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% binomial_fact_lemma
thf(fact_2750_choose__mult,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2751_binomial__absorb__comp,axiom,
! [N2: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_2752_binomial__Suc__Suc__eq__times,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_2753_nat__abs__mult__distrib,axiom,
! [W: int,Z2: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W @ Z2 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W ) ) @ ( nat2 @ ( abs_abs @ int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_2754_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).
% mask_Suc_double
thf(fact_2755_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times @ num @ X @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_2756_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_2757_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_2758_binomial__absorption,axiom,
! [K: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2759_nat__mult__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_2760_binomial__altdef__nat,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2761_times__binomial__minus1__eq,axiom,
! [K: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2762_nat__mult__distrib__neg,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_2763_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2764_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2765_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_2766_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_2767_choose__two,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_2768_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N4: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2769_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K5: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K5 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_2770_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N4: nat,K5: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N4 ) @ K5 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K5 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_2771_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_2772_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M4: nat,N4: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M4
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_2773_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_2774_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_2775_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_2776_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_2777_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_2778_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_2779_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_2780_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_2781_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_2782_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A4 )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_2783_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_2784_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% push_bit_of_1
thf(fact_2785_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N4: nat,A8: A] : ( bit_se5824344971392196577ns_xor @ A @ A8 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_2786_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A8 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_2787_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N2 )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_2788_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N2 ) ) ) ).
% not_bit_1_Suc
thf(fact_2789_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% bit_numeral_simps(1)
thf(fact_2790_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N4: nat,A8: A] : ( bit_se1065995026697491101ons_or @ A @ A8 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_2791_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N4: nat,K5: int] : ( times_times @ int @ K5 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% push_bit_int_def
thf(fact_2792_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A4: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_2793_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N4: nat,M4: nat] : ( times_times @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% push_bit_nat_def
thf(fact_2794_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N4: nat,A8: A] : ( times_times @ A @ A8 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_2795_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K5: int] :
( plus_plus @ int @ K5
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K5 @ N4 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% set_bit_eq
thf(fact_2796_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N4: nat,K5: int] : ( minus_minus @ int @ K5 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K5 @ N4 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_2797_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2798_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N2: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2799_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,B3: A,N2: nat] :
( ! [J4: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A4 @ ( suc @ J4 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N2 )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N2 ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2800_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M4: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_2801_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M4: nat,N4: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M4
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_2802_take__bit__Suc__from__most,axiom,
! [N2: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2803_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K5: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K5 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_2804_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M4: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_2805_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_2806_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K5: int,L2: int] :
( if @ int
@ ( K5
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K5 )
@ ( if @ int
@ ( K5
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K5
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_2807_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K5: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K5 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_2808_card__UNION,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A3 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ( finite_finite2 @ 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_2809_or__not__numerals_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_2810_or__not__numerals_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_2811_and__not__numerals_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_2812_image__ident,axiom,
! [A: $tType,Y5: set @ A] :
( ( image2 @ A @ A
@ ^ [X2: A] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_2813_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,P: B > $o] :
( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y3: A] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_2814_vimage__ident,axiom,
! [A: $tType,Y5: set @ A] :
( ( vimage @ A @ A
@ ^ [X2: A] : X2
@ Y5 )
= Y5 ) ).
% vimage_ident
thf(fact_2815_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [Y3: B] : Y3 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_2816_bool__assn__proper_I4_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H4 )
& ( Q2 @ H4 ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_2817_bool__assn__proper_I3_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H4 )
| ( Q2 @ H4 ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_2818_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ).
% bool_assn_proper(2)
thf(fact_2819_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_2820_singleton__conv,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A4 )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_2821_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_2822_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_2823_Field__square,axiom,
! [A: $tType,X: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X
@ ^ [Uu: A] : X ) )
= X ) ).
% Field_square
thf(fact_2824_bool__assn__proper_I5_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H4 )
& ~ ( P @ H4 ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_2825_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_2826_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B5 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_2827_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B5 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_2828_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_2829_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
@ ^ [I4: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_2830_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
@ ^ [I4: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_2831_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_2832_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A4 ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2833_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( A4 = K5 ) @ ( B3 @ K5 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A4 ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( A4 = K5 ) @ ( B3 @ K5 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2834_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: B > A,G: B > A,S: set @ B] :
( ( image2 @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ S )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ S @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S
@ ( collect @ B
@ ^ [X2: B] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2835_UN__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_2836_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A3: set @ B] :
( ( ( member @ B @ C2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C2
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_2837_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_2838_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_2839_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A3 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_2840_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A3 )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_2841_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_2842_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_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_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 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ A3 )
& ( ord_less @ A @ ( F2 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_2851_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_2852_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ B,A4: A,B5: B > ( set @ A )] :
( ( ( C6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A4 @ ( B5 @ X2 ) )
@ C6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A4 @ ( B5 @ X2 ) )
@ C6 ) )
= ( insert2 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2853_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_2854_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_2855_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ C,A3: C > ( set @ D ),B5: set @ D] :
( ( ( C6
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C6 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2856_UN__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C6: set @ F3,A3: set @ E,B5: F3 > ( set @ E )] :
( ( ( C6
= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) )
= ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B5 @ C6 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2857_UN__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A4: B,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B5 @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% UN_insert
thf(fact_2858_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_2859_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_2860_INT__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A3 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_2861_INT__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A4: B,A3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( insert2 @ B @ A4 @ A3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B5 @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% INT_insert
thf(fact_2862_fst__image__times,axiom,
! [B: $tType,A: $tType,B5: set @ B,A3: set @ A] :
( ( ( B5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
= A3 ) ) ) ).
% fst_image_times
thf(fact_2863_snd__image__times,axiom,
! [B: $tType,A: $tType,A3: set @ B,B5: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B5 ) )
= ( 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] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_2864_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A3: set @ B,D3: B,B5: set @ A] :
( ( ( member @ B @ C2 @ A3 )
=> ( ( ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B5 ) @ C2 @ D3 )
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B5 ) @ C2 @ D3 )
@ A3 )
= B5 ) ) ) )
& ( ~ ( member @ B @ C2 @ A3 )
=> ( ( ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B5 ) @ C2 @ D3 )
@ A3 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) ) )
& ( ~ ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B5 ) @ C2 @ D3 )
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_2865_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A3: set @ A,X4: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : X4 )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_2866_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B5: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F2 )
@ ( product_Sigma @ B @ B @ A3
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F2 @ A3 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F2 @ B5 ) ) ) ).
% pairself_image_cart
thf(fact_2867_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_2868_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C6: set @ A,A3: set @ B,B5: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : A3 )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : B5 ) )
= ( ( C6
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A3 @ B5 ) ) ) ).
% disjnt_Times1_iff
thf(fact_2869_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,C6: set @ B,B5: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) )
= ( ( C6
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A3 @ B5 ) ) ) ).
% disjnt_Times2_iff
thf(fact_2870_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2871_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o,F2: B > A] :
( ( finite_finite2 @ 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_2872_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A,P: B > $o] :
( ( finite_finite2 @ 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_2873_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ A,A3: A > ( set @ B ),B5: set @ B] :
( ( ( C6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C6 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2874_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ D,A3: set @ C,B5: D > ( set @ C )] :
( ( ( C6
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) )
= ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C6 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2875_INT__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C6: set @ E,A3: E > ( set @ F3 ),B5: set @ F3] :
( ( ( C6
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) )
= ( top_top @ ( set @ F3 ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) )
= ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A3 @ C6 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2876_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,B5: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A4 @ A3 )
@ ^ [Uu: A] : ( insert2 @ B @ B3 @ B5 ) )
= ( 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 @ B5 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A4 @ A3 )
@ ^ [Uu: A] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2877_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_2878_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C,A3: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F2 @ A3 ) ) ).
% inj_on_apfst
thf(fact_2879_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: B > C,A3: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A3 ) )
= ( inj_on @ B @ C @ F2 @ A3 ) ) ).
% inj_on_apsnd
thf(fact_2880_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2881_INT__simps_I4_J,axiom,
! [G2: $tType,H8: $tType,C6: set @ H8,A3: set @ G2,B5: H8 > ( set @ G2 )] :
( ( ( C6
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H8 @ ( set @ G2 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) )
= ( top_top @ ( set @ G2 ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H8 @ ( set @ G2 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) )
= ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H8 @ ( set @ G2 ) @ B5 @ C6 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2882_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R4 )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S ) ) ).
% pred_subset_eq2
thf(fact_2883_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A5: set @ A] :
( collect @ ( set @ A )
@ ^ [B8: set @ A] : ( ord_less_eq @ ( set @ A ) @ B8 @ A5 ) ) ) ) ).
% Pow_def
thf(fact_2884_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B8 ) ) ) ) ).
% less_set_def
thf(fact_2885_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B5: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_2886_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_2887_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_2888_pred__subset__eq,axiom,
! [A: $tType,R4: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R4 )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R4 @ S ) ) ).
% pred_subset_eq
thf(fact_2889_prop__restrict,axiom,
! [A: $tType,X: A,Z6: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X @ Z6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z6
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_2890_Collect__restrict,axiom,
! [A: $tType,X4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_2891_pairwise__image,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F2: B > A,S2: set @ B] :
( ( pairwise @ A @ R2 @ ( image2 @ B @ A @ F2 @ S2 ) )
= ( pairwise @ B
@ ^ [X2: B,Y3: B] :
( ( ( F2 @ X2 )
!= ( F2 @ Y3 ) )
=> ( R2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
@ S2 ) ) ).
% pairwise_image
thf(fact_2892_inv__imagep__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_imagep @ B @ A )
= ( ^ [R5: B > B > $o,F4: A > B,X2: A,Y3: A] : ( R5 @ ( F4 @ X2 ) @ ( F4 @ Y3 ) ) ) ) ).
% inv_imagep_def
thf(fact_2893_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A5: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2894_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_2895_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A5: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_2896_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% minus_set_def
thf(fact_2897_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ~ ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2898_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_2899_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X2: B] : ( F2 @ ( G @ X2 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F2 @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_2900_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B8: set @ B] :
( collect @ A
@ ^ [X2: A] : ( member @ B @ ( F4 @ X2 ) @ B8 ) ) ) ) ).
% vimage_def
thf(fact_2901_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C6: set @ A,A3: set @ B,B5: set @ B] :
( ( member @ A @ X @ C6 )
=> ( ( ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : C6 )
= ( product_Sigma @ B @ A @ B5
@ ^ [Uu: B] : C6 ) )
= ( A3 = B5 ) ) ) ).
% Times_eq_cancel2
thf(fact_2902_Sigma__cong,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,C6: A > ( set @ B ),D4: A > ( set @ B )] :
( ( A3 = B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( C6 @ X3 )
= ( D4 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A3 @ C6 )
= ( product_Sigma @ A @ B @ B5 @ D4 ) ) ) ) ).
% Sigma_cong
thf(fact_2903_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q2 @ X2 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_2904_Int__Collect,axiom,
! [A: $tType,X: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_2905_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% Int_def
thf(fact_2906_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% inf_set_def
thf(fact_2907_inf__Int__eq,axiom,
! [A: $tType,R4: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R4 )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ R4 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_2908_pairwise__trivial,axiom,
! [A: $tType,I: set @ A] :
( pairwise @ A
@ ^ [I4: A,J3: A] : J3 != I4
@ I ) ).
% pairwise_trivial
thf(fact_2909_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C2: C] :
( ( product_curry @ A @ B @ C
@ ^ [X2: product_prod @ A @ B] : C2 )
= ( ^ [X2: A,Y3: B] : C2 ) ) ).
% curry_K
thf(fact_2910_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q2 @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_2911_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
| ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% Un_def
thf(fact_2912_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% sup_set_def
thf(fact_2913_sup__Un__eq,axiom,
! [A: $tType,R4: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R4 )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R4 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_2914_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_2915_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image2 @ B @ A @ F2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_2916_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A3: set @ C] :
( ( image2 @ B @ A @ F2 @ ( image2 @ C @ B @ G @ A3 ) )
= ( image2 @ C @ A
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_2917_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,A3: set @ B] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F2 @ X3 ) )
=> ~ ( member @ B @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_2918_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_2919_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_2920_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A3: set @ A,G: C > B,B5: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A3 ) ) ) ).
% sum_product
thf(fact_2921_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,A3: set @ B,R2: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ R2 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ ( F2 @ N4 ) @ R2 )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_2922_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R2: A,F2: B > A,A3: set @ B] :
( ( times_times @ A @ R2 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ R2 @ ( F2 @ N4 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_2923_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_2924_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H4: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_2925_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R4 )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_2926_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B6: A] :
( ( B6 != A8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A8 ) @ R5 ) ) ) ) ) ).
% underS_def
thf(fact_2927_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 )
@ ^ [I4: set @ ( product_prod @ A @ B ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ I4 )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2928_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R4 ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_2929_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A8 ) @ R5 ) ) ) ) ).
% under_def
thf(fact_2930_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_2931_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_2932_insert__Collect,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( insert2 @ A @ A4 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U3: A] :
( ( U3 != A4 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_2933_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A8: A,B8: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A8 )
| ( member @ A @ X2 @ B8 ) ) ) ) ) ).
% insert_compr
thf(fact_2934_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% Set.empty_def
thf(fact_2935_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B,C6: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 )
= ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : D4 ) )
= ( ( ( A3 = C6 )
& ( B5 = D4 ) )
| ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C6
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_2936_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A8: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A8 ) ) ) ) ).
% insert_def
thf(fact_2937_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_2938_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_2939_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_2940_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B5: set @ I6,F5: I6 > ( filter @ A )] :
( ! [X8: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X8 @ B5 )
=> ( ( finite_finite2 @ I6 @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F5 @ X8 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F5 @ B5 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_2941_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X4: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( F2 @ X2 ) )
@ X4 ) ).
% inj_on_convol_ident
thf(fact_2942_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( C2 @ X2 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_2943_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X2: A] : ( product_Pair @ B @ A @ ( C2 @ X2 ) @ X2 )
@ S ) ).
% inj_Pair(2)
thf(fact_2944_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C5: ( product_prod @ A @ B ) > C,X2: A,Y3: B] : ( C5 @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) ) ).
% curry_def
thf(fact_2945_SUP__Sup__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X2: A] : ( member @ A @ X2 @ I4 )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_2946_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 )
@ ^ [I4: set @ ( product_prod @ A @ B ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ I4 )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% INF_Int_eq2
thf(fact_2947_INF__Int__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X2: A] : ( member @ A @ X2 @ I4 )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_2948_bot__assn__def,axiom,
( ( bot_bot @ assn )
= ( abs_assn
@ ^ [Uu: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ) ).
% bot_assn_def
thf(fact_2949_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I )
& ( ( X @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I )
& ( ( times_times @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2950_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C2
@ A3 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_2951_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A3: set @ A,C2: B] :
( ( member @ A @ X @ A3 )
=> ( ( image2 @ A @ B
@ ^ [X2: A] : C2
@ A3 )
= ( insert2 @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_2952_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I4 ) )
@ I )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I ) ) ) ) ).
% sum_power_add
thf(fact_2953_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_2954_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A8: B] : ( member @ A @ ( F2 @ A8 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_2955_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_2956_UN__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_2957_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B5 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_2958_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B5 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_2959_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set @ A,A4: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( insert2 @ B @ A4 @ ( B5 @ X2 ) )
@ A3 ) )
= ( insert2 @ B @ A4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2960_vimage__fst,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
= ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_2961_vimage__snd,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A3 ) ) ).
% vimage_snd
thf(fact_2962_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C6: set @ A,A3: set @ B,B5: set @ B] :
( ( member @ A @ X @ C6 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : C6 )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu: B] : C6 ) )
= ( ord_less_eq @ ( set @ B ) @ A3 @ B5 ) ) ) ).
% Times_subset_cancel2
thf(fact_2963_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: set @ A,B5: set @ B] :
( ( member @ A @ ( product_fst @ A @ B @ X ) @ A3 )
=> ( ( member @ B @ ( product_snd @ A @ B @ X ) @ B5 )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_2964_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: set @ A,B5: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A3 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_2965_R__subset__Field,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R4 )
@ ^ [Uu: A] : ( field2 @ A @ R4 ) ) ) ).
% R_subset_Field
thf(fact_2966_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set @ A,A4: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A3 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( insert2 @ B @ A4 @ ( B5 @ X2 ) )
@ A3 ) )
= ( insert2 @ B @ A4 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2967_INT__extend__simps_I5_J,axiom,
! [I6: $tType,J5: $tType,A4: I6,B5: J5 > ( set @ I6 ),C6: set @ J5] :
( ( insert2 @ I6 @ A4 @ ( complete_Inf_Inf @ ( set @ I6 ) @ ( image2 @ J5 @ ( set @ I6 ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ I6 )
@ ( image2 @ J5 @ ( set @ I6 )
@ ^ [X2: J5] : ( insert2 @ I6 @ A4 @ ( B5 @ X2 ) )
@ C6 ) ) ) ).
% INT_extend_simps(5)
thf(fact_2968_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I: set @ A,X4: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I @ X4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ I )
=> ( ( X4 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_2969_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B5: C > ( set @ B ),A3: set @ C] :
( ( image @ B @ A @ R2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R2 @ ( B5 @ X2 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_2970_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [F: A > nat] :
( ? [N3: nat] :
( ( image2 @ A @ nat @ F @ A3 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) )
=> ~ ( inj_on @ A @ nat @ F @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_2971_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B,C6: set @ A,D4: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : D4 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A3 @ C6 )
@ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B5 @ D4 ) ) ) ).
% Times_Int_Times
thf(fact_2972_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I
@ ^ [I4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ I4 ) @ ( B5 @ I4 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A3 ) @ ( product_Sigma @ A @ B @ I @ B5 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_2973_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A3 @ B5 )
@ ^ [Uu: A] : C6 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Int_distrib1
thf(fact_2974_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I
@ ^ [I4: A] : ( sup_sup @ ( set @ B ) @ ( A3 @ I4 ) @ ( B5 @ I4 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A3 ) @ ( product_Sigma @ A @ B @ I @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_2975_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B5 )
@ ^ [Uu: A] : C6 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Un_distrib1
thf(fact_2976_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X2: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X2 )
= ( product_snd @ A @ A @ X2 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_2977_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A3: set @ A,B5: set @ B,S2: set @ ( product_prod @ B @ C ),C6: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S2
@ ( product_Sigma @ B @ C @ B5
@ ^ [Uu: B] : C6 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R2 @ S2 )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu: A] : C6 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_2978_Sigma__Union,axiom,
! [B: $tType,A: $tType,X4: set @ ( set @ A ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X4 ) @ B5 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A5: set @ A] : ( product_Sigma @ A @ B @ A5 @ B5 )
@ X4 ) ) ) ).
% Sigma_Union
thf(fact_2979_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A3 @ B5 )
@ ^ [Uu: A] : C6 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Diff_distrib1
thf(fact_2980_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I
@ ^ [I4: A] : ( minus_minus @ ( set @ B ) @ ( A3 @ I4 ) @ ( B5 @ I4 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A3 ) @ ( product_Sigma @ A @ B @ I @ B5 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_2981_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R4 )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_2982_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S7: set @ A] :
( ( minus_minus @ ( set @ A ) @ S7 @ S )
= ( minus_minus @ ( set @ A ) @ S3 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2983_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_2984_Id__on__subset__Times,axiom,
! [A: $tType,A3: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id_on @ A @ A3 )
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) ) ).
% Id_on_subset_Times
thf(fact_2985_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X2: A] : ( member @ A @ X2 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_2986_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X2: A] : ( member @ A @ X2 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_2987_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: C] : ( converse @ B @ A @ ( R2 @ X2 ) )
@ S ) ) ) ).
% converse_UNION
thf(fact_2988_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S2: set @ ( product_prod @ A @ C ),R2: D > ( set @ ( product_prod @ C @ B ) ),I: set @ D] :
( ( relcomp @ A @ C @ B @ S2 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R2 @ I ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ S2 @ ( R2 @ I4 ) )
@ I ) ) ) ).
% relcomp_UNION_distrib
thf(fact_2989_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R2: D > ( set @ ( product_prod @ A @ C ) ),I: set @ D,S2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R2 @ I ) ) @ S2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ ( R2 @ I4 ) @ S2 )
@ I ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_2990_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_2991_product__swap,axiom,
! [B: $tType,A: $tType,A3: set @ B,B5: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : A3 ) ) ).
% product_swap
thf(fact_2992_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: C] : ( converse @ B @ A @ ( R2 @ X2 ) )
@ S ) ) ) ).
% converse_INTER
thf(fact_2993_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_2994_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A3 ) )
= ( bot_bot @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_2995_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_2996_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_2997_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A3 ) )
= ( top_top @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_2998_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: set @ B,X: A,F2: B > A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X @ ( F2 @ I4 ) )
@ I ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_2999_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: set @ B,F2: B > A,X: A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F2 @ I4 ) @ X )
@ I ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_3000_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
= ( times_times @ A @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_even
thf(fact_3001_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_3002_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: B,A3: set @ B] :
( ( 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 ) ) ) ) ) ).
% INF_insert
thf(fact_3003_sup__assn__def,axiom,
( ( sup_sup @ assn )
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H4 )
| ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% sup_assn_def
thf(fact_3004_inf__assn__def,axiom,
( ( inf_inf @ assn )
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H4 )
& ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% inf_assn_def
thf(fact_3005_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
= ( times_times @ A @ ( times_times @ A @ Z2 @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_odd
thf(fact_3006_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B5 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_restrict
thf(fact_3007_Image__singleton,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A4: B] :
( ( image @ B @ A @ R2 @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B6: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B6 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_3008_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3009_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ B,A4: A,B5: B > ( set @ A )] :
( ( ( C6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) )
= ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A4 @ ( B5 @ X2 ) )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_3010_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C6: set @ C,A3: C > ( set @ D ),B5: set @ D] :
( ( ( C6
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C6 ) ) @ B5 )
= B5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C6 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3011_UN__extend__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C6: set @ F3,A3: set @ E,B5: F3 > ( set @ E )] :
( ( ( C6
= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B5 @ C6 ) ) )
= A3 ) )
& ( ( C6
!= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B5 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3012_times__subset__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,C6: set @ B,B5: set @ A,D4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : D4 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( C6
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
& ( ord_less_eq @ ( set @ B ) @ C6 @ D4 ) ) ) ) ).
% times_subset_iff
thf(fact_3013_INT__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_3014_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C6: set @ A,A3: A > ( set @ B ),B5: set @ B] :
( ( ( C6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C6 ) ) @ B5 )
= B5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C6 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3015_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ D,A3: set @ C,B5: D > ( set @ C )] :
( ( ( C6
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C6 ) ) )
= A3 ) )
& ( ( C6
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3016_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) )
=> ( ( A4 = B3 )
| ( member @ A @ A4 @ A3 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_3017_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I: set @ A,F2: B > C,A3: A > ( set @ B )] :
( ( I
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ( inj_on @ B @ C @ F2 @ ( A3 @ I3 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ I ) ) ) ) ) ).
% inj_on_INTER
thf(fact_3018_wfI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : B5 ) )
=> ( ! [X3: A,P4: A > $o] :
( ! [Xa2: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R2 )
=> ( P4 @ Y2 ) )
=> ( P4 @ Xa2 ) )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ X3 @ B5 )
=> ( P4 @ X3 ) ) ) )
=> ( wf @ A @ R2 ) ) ) ).
% wfI
thf(fact_3019_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A3 )
& ( finite_finite2 @ B @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_3020_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_3021_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_cartesian_productD1
thf(fact_3022_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( B5 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( finite_finite2 @ B @ ( B5 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_3023_quotient__diff1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A8: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ R2 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A3 @ R2 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) ) ) ) ) ).
% quotient_diff1
thf(fact_3024_Image__subset,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),A3: set @ A,B5: set @ B,C6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R2 @ C6 ) @ B5 ) ) ).
% Image_subset
thf(fact_3025_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W: A,E3: set @ ( product_prod @ A @ A ),U2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ U2
@ ^ [Uu: A] : U2 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_rtrancl @ A @ E3 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_3026_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set @ A,T3: set @ B,R4: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T3 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R4 @ I4 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T3 )
& ( R4 @ I4 @ J3 ) ) ) )
@ S )
= ( times_times @ nat @ K @ ( finite_card @ B @ T3 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3027_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B5: C > ( set @ B ),A3: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A3 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R2 @ ( B5 @ X2 ) )
@ A3 ) ) ) ).
% Image_INT_subset
thf(fact_3028_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A8: nat,B6: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_3029_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B5 ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( B5 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_3030_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W: A,E3: set @ ( product_prod @ A @ A ),U2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ U2
@ ^ [Uu: A] : U2 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E3 ) ) ) ).
% Restr_trancl_mono
thf(fact_3031_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A5: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_3032_refl__onI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( 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 ) @ R2 ) )
=> ( refl_on @ A @ A3 @ R2 ) ) ) ).
% refl_onI
thf(fact_3033_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_3034_homo__rel__restrict__mono,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),B5: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu: A] : B5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R4 @ A3 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A3 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_3035_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 )
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_3036_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: B > A,A3: set @ B,A10: set @ A,G: D > C,B5: set @ D,B12: set @ C] :
( ( ( image2 @ B @ A @ F2 @ A3 )
= A10 )
=> ( ( ( image2 @ D @ C @ G @ B5 )
= B12 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F2 @ G )
@ ( product_Sigma @ B @ D @ A3
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ C @ A10
@ ^ [Uu: A] : B12 ) ) ) ) ).
% map_prod_surj_on
thf(fact_3037_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A5 )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_3038_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) )
= ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B5 ) ) ) ).
% card_cartesian_product
thf(fact_3039_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F2 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_3040_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B10: set @ ( set @ A ),A3: set @ A] :
( ( ( B10
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B10 ) )
= A3 ) )
& ( ( B10
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B10 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B10 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_3041_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B10: set @ ( set @ A ),A3: set @ A] :
( ( ( B10
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B10 ) @ A3 )
= A3 ) )
& ( ( B10
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B10 ) @ A3 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B8: set @ A] : ( inf_inf @ ( set @ A ) @ B8 @ A3 )
@ B10 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_3042_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F2: A > B,A3: set @ A,G: C > D,B5: set @ C] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( inj_on @ C @ D @ G @ B5 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F2 @ G )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu: A] : B5 ) ) ) ) ).
% map_prod_inj_on
thf(fact_3043_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_3044_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_3045_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_3046_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,G: A > B,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( inj_on @ A @ B @ G @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A3 ) @ ( image2 @ A @ B @ G @ B5 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A3 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_3047_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [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_3048_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [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_3049_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_3050_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_3051_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A3 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A8: B] : ( sup_sup @ A @ ( F2 @ A8 ) @ ( G @ A8 ) )
@ A3 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3052_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A3 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A8: B] : ( inf_inf @ A @ ( F2 @ A8 ) @ ( G @ A8 ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3053_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),B8: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( image @ B @ A @ R5 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B8 ) ) ) ) ).
% Image_eq_UN
thf(fact_3054_INT__extend__simps_I3_J,axiom,
! [F3: $tType,E: $tType,C6: set @ E,A3: E > ( set @ F3 ),B5: set @ F3] :
( ( ( C6
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A3 @ C6 ) ) @ B5 )
= ( minus_minus @ ( set @ F3 ) @ ( top_top @ ( set @ F3 ) ) @ B5 ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A3 @ C6 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A3 @ X2 ) @ B5 )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_3055_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_3056_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B8: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( vimage @ A @ B @ F4 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B8 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3057_INT__extend__simps_I4_J,axiom,
! [G2: $tType,H8: $tType,C6: set @ H8,A3: set @ G2,B5: H8 > ( set @ G2 )] :
( ( ( C6
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H8 @ ( set @ G2 ) @ B5 @ C6 ) ) )
= A3 ) )
& ( ( C6
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H8 @ ( set @ G2 ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H8 @ ( set @ G2 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B5 @ X2 ) )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_3058_Sigma__Image,axiom,
! [A: $tType,B: $tType,A3: set @ B,B5: B > ( set @ A ),X4: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A3 @ B5 ) @ X4 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ B ) @ X4 @ A3 ) ) ) ) ).
% Sigma_Image
thf(fact_3059_Domain__Union,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S ) ) ) ).
% Domain_Union
thf(fact_3060_rtrancl__last__visit_H,axiom,
! [A: $tType,Q4: A,Q5: A,R4: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q5 ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( 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 @ R4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_3061_rtrancl__last__touch,axiom,
! [A: $tType,Q4: A,Q5: A,R4: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q5 ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( ( member @ A @ Q4 @ S )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_3062_Range__Union,axiom,
! [A: $tType,B: $tType,S: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range2 @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range2 @ B @ A ) @ S ) ) ) ).
% Range_Union
thf(fact_3063_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( image2 @ A @ B @ F2 @ A3 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K5: A] : ( insert2 @ B @ ( F2 @ K5 ) )
@ ( bot_bot @ ( set @ B ) )
@ A3 ) ) ) ).
% image_fold_insert
thf(fact_3064_product__fold,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: A,Z3: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
@ Z3
@ B5 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A3 ) ) ) ) ).
% product_fold
thf(fact_3065_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A3: set @ B,B5: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A3 @ B5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% snd_image_Sigma
thf(fact_3066_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 )
@ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) ) ).
% subset_fst_snd
thf(fact_3067_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A3: set @ A,R4: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu: A] : A3 ) )
@ R4 )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A3 @ R4 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ A3 @ R4 ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_3068_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_3069_Id__on__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ 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_3070_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X4: C > ( set @ ( product_prod @ B @ A ) ),I: set @ C,S: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X4 @ I ) ) @ S )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I4: C] : ( image @ B @ A @ ( X4 @ I4 ) @ S )
@ I ) ) ) ).
% UN_Image
thf(fact_3071_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M4: nat,N4: nat] :
( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K5: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K5 @ N4 ) @ M4 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_3072_Chains__alt__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ( chains @ A @ R2 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_3073_Chains__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R2 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) ) ) ).
% Chains_subset
thf(fact_3074_uminus__assn__def,axiom,
( ( uminus_uminus @ assn )
= ( ^ [P2: assn] :
( abs_assn
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H4 )
& ~ ( rep_assn @ P2 @ H4 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_3075_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_3076_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_3077_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( C2 @ K5 ) )
@ S )
= ( plus_plus @ A @ ( B3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( C2 @ K5 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3078_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( C2 @ K5 ) )
@ S )
= ( times_times @ A @ ( B3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( C2 @ K5 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3079_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I )
=> ( finite_finite2 @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X2 ) )
@ I ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_3080_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I )
=> ( finite_finite2 @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X2 ) )
@ I ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_3081_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B5: C > ( set @ D ),F2: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( B5 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F2 @ ( B5 @ X2 ) )
@ A3 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B5 @ A3 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( B5 @ X2 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_3082_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B5: C > ( set @ D ),F2: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( B5 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F2 @ ( B5 @ X2 ) )
@ A3 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B5 @ A3 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( B5 @ X2 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_3083_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A13: set @ A,R1: set @ ( product_prod @ A @ A ),A23: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A13 @ R1 )
=> ( ( equiv_equiv @ B @ A23 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F2 )
=> ( ( member @ A @ A1 @ A13 )
=> ( ( member @ B @ A22 @ A23 )
=> ( ( 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_3084_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E3 @ S ) @ S )
=> ( ( member @ A @ U @ S )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ S
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_3085_rtrancl__last__visit,axiom,
! [A: $tType,Q4: A,Q5: A,R4: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q5 ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( 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 @ R4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_3086_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E3: set @ ( product_prod @ A @ A ),R4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : R4 ) ) ) )
=> ( ~ ( member @ A @ U @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E3 @ R4 ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_3087_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),A3: set @ C,B5: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R2 @ ( B5 @ X2 ) )
@ A3 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_3088_card__quotient__disjoint,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A3 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A3 @ R2 ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_3089_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A5: set @ A,B8: 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 ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B8 @ X2 ) ) )
@ A5 ) ) ) ) ).
% Sigma_def
thf(fact_3090_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A5: set @ A,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R5 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ) ).
% quotient_def
thf(fact_3091_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A8: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A8 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_3092_Chains__subset_H,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) ) )
@ ( chains @ A @ R2 ) ) ) ).
% Chains_subset'
thf(fact_3093_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A8: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A8 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_3094_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A8: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A8 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_3095_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A5: 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 ) ) ) )
@ A5 ) ) ) ) ).
% Id_on_def
thf(fact_3096_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I: set @ A,X: A > B,A4: A > B,B3: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ( 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
@ ^ [I4: A] : ( times_times @ B @ ( A4 @ I4 ) @ ( X @ I4 ) )
@ I )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3097_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N2 ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3098_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R4: set @ B,G: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ S ) @ R4 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y3: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
& ( ( G @ X2 )
= Y3 ) ) ) ) )
@ ( F2 @ Y3 ) )
@ R4 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3099_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C2: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ C2 )
@ S )
= ( times_times @ A @ ( B3 @ A4 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ C2 )
@ S )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3100_Gcd__eq__Max,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( M2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( gcd_Gcd @ nat @ M2 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M4: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M4 ) )
@ M2 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_3101_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I )
=> ( finite_finite2 @ B @ ( A3 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I )
=> ( ( 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 @ I ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A3 @ I4 ) )
@ I ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_3102_trancl__multi__insert2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),M: A,X4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( insert2 @ A @ M @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : X4 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ M ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_3103_trancl__multi__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),X4: set @ A,M: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ X4
@ ^ [Uu: A] : ( insert2 @ A @ M @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_3104_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set @ A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B5 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B5 ) )
@ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) )
@ A3
@ B5 ) ) ) ).
% fold_union_pair
thf(fact_3105_wf__UN,axiom,
! [B: $tType,A: $tType,I: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ( wf @ B @ ( R2 @ I3 ) ) )
=> ( ! [I3: A,J4: A] :
( ( member @ A @ I3 @ I )
=> ( ( member @ A @ J4 @ I )
=> ( ( ( R2 @ I3 )
!= ( R2 @ J4 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R2 @ I3 ) ) @ ( range2 @ B @ B @ ( R2 @ J4 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ I ) ) ) ) ) ).
% wf_UN
thf(fact_3106_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > ( product_prod @ B @ C ),A3: set @ B,B5: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F2
@ ( product_Sigma @ B @ C @ A3
@ ^ [Uu: B] : B5 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 ) @ A3 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) @ B5 ) ) ) ).
% vimage_Times
thf(fact_3107_rtrancl__last__visit__node,axiom,
! [A: $tType,S2: A,S3: A,R4: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ S3 ) @ ( transitive_rtrancl @ A @ R4 ) )
=> ( ( ( S2 != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ S3 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( 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 @ R4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S3 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( 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_3108_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A8: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A8 ) @ N4 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A8 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N4 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A8 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3109_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > A,G: C > B,A3: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A3 )
@ ^ [X2: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3110_trans__wf__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( wf @ A @ R2 )
= ( ! [A8: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu: A] : ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_3111_range__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F2 @ ( top_top @ ( set @ C ) ) )
@ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F2 ) @ ( top_top @ ( set @ C ) ) )
@ ^ [Uu: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F2 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_3112_Pow__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( pow2 @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A5: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A5 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A5 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A3 ) ) ) ).
% Pow_fold
thf(fact_3113_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A8: A,N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A8 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3114_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N4: nat,A8: A] : ( bit_se5824344872417868541ns_and @ A @ A8 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_3115_and__not__numerals_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_3116_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_3117_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K5: nat] :
( if @ A
@ ( K5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A8 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K5 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K5 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3118_and__not__numerals_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_3119_and__not__numerals_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_3120_or__not__numerals_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_3121_or__not__numerals_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_3122_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A8: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N4 @ A8 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A8 @ N4 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N4 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_3123_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N2: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K5: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K5 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3124_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R4 ) )
= ( same_fst @ A @ B @ P
@ ^ [X2: A] : ( transitive_trancl @ B @ ( R4 @ X2 ) ) ) ) ).
% same_fst_trancl
thf(fact_3125_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X2: B] : C2
@ A3 ) )
= C2 ) ) ).
% subset_mset.cSUP_const
thf(fact_3126_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X2: B] : C2
@ A3 ) )
= C2 ) ) ).
% subset_mset.cINF_const
thf(fact_3127_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_even_sum
thf(fact_3128_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_odd_sum
thf(fact_3129_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q2: A > B > $o] :
( ! [X3: A,Y2: B] :
( ( P @ X3 @ Y2 )
=> ( Q2 @ X3 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 ) ) ).
% predicate2I
thf(fact_3130_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_3131_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_3132_predicate1I,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q2 ) ) ).
% predicate1I
thf(fact_3133_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_3134_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_3135_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_3136_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_3137_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_3138_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A4 = B3 )
& ( B3 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_3139_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_3140_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3141_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3142_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_3143_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3144_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_3145_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_3146_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_3147_Inf__filter__not__bot,axiom,
! [A: $tType,B5: set @ ( filter @ A )] :
( ! [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B5 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X8 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B5 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_3148_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I: set @ A,F5: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ! [J4: A] :
( ( member @ A @ J4 @ I )
=> ? [X6: A] :
( ( member @ A @ X6 @ I )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X6 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I3 ) @ ( F5 @ J4 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ I )
& ( ( F5 @ X2 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_3149_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q2: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
=> ( Q2 @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_3150_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q2 )
=> ( Q2 @ X ) ) ) ).
% rev_predicate1D
thf(fact_3151_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q2: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
=> ( ( P @ X @ Y )
=> ( Q2 @ X @ Y ) ) ) ).
% predicate2D
thf(fact_3152_predicate1D,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q2 )
=> ( ( P @ X )
=> ( Q2 @ X ) ) ) ).
% predicate1D
thf(fact_3153_refl__ge__eq,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [X3: A] : ( R4 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ R4 ) ) ).
% refl_ge_eq
thf(fact_3154_ge__eq__refl,axiom,
! [A: $tType,R4: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ R4 )
=> ( R4 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_3155_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_3156_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_3157_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N2: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3158_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3159_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_3160_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_3161_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3162_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3163_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3164_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3165_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3166_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
@ ^ [A8: nat] : ( times_times @ A @ ( F2 @ A8 ) )
@ A4
@ B3
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3167_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P3 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3168_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ K5 ) ) @ K5 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ).
% gbinomial_parallel_sum
thf(fact_3169_vandermonde,axiom,
! [M: nat,N2: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K5: nat] : ( times_times @ nat @ ( binomial @ M @ K5 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R2 @ K5 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N2 ) @ R2 ) ) ).
% vandermonde
thf(fact_3170_fact__eq__fact__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3171_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3172_binomial,axiom,
! [A4: nat,B3: nat,N2: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A4 @ B3 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K5: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K5 ) ) @ ( power_power @ nat @ A4 @ K5 ) ) @ ( power_power @ nat @ B3 @ ( minus_minus @ nat @ N2 @ K5 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ).
% binomial
thf(fact_3173_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_3174_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_3175_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( 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 ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A4 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_3176_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( 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 ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A4 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A4 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_3177_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( 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 ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A4 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A4 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_3178_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( 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 ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_3179_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.in_pairs_0
thf(fact_3180_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3181_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( gbinomial @ A @ A4 @ K5 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K5 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_3182_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,B3: A,N2: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A4 @ B3 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K5 ) ) @ ( power_power @ A @ A4 @ K5 ) ) @ ( power_power @ A @ B3 @ ( minus_minus @ nat @ N2 @ K5 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% binomial_ring
thf(fact_3183_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N2: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3184_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.in_pairs
thf(fact_3185_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.in_pairs
thf(fact_3186_choose__square__sum,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K5: nat] : ( power_power @ nat @ ( binomial @ N2 @ K5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ).
% choose_square_sum
thf(fact_3187_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A4: A,B3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A4 @ B3 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K5 ) ) @ ( comm_s3205402744901411588hammer @ A @ A4 @ K5 ) ) @ ( comm_s3205402744901411588hammer @ A @ B3 @ ( minus_minus @ nat @ N2 @ K5 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3188_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 )
=> ( ( ord_less_eq @ nat @ K @ P3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3189_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A4: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A4 ) @ K5 ) @ ( power_power @ A @ X @ K5 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K5 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A4 ) @ K5 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K5 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K5 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3190_gauss__sum__nat,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_3191_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3192_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3193_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A4: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A4 ) @ K5 ) @ ( power_power @ A @ X @ K5 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K5 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K5 ) @ A4 ) @ ( one_one @ A ) ) @ K5 ) @ ( power_power @ A @ X @ K5 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K5 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3194_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( N2
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3195_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_3196_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,D3: A,N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_3197_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_3198_choose__linear__sum,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ I4 @ ( binomial @ N2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_3199_arith__series__nat,axiom,
! [A4: nat,D3: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A4 @ ( times_times @ nat @ I4 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A4 ) @ ( times_times @ nat @ N2 @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3200_Sum__Icc__nat,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_3201_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3202_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: 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 ) ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_3203_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_3204_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,D3: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3205_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3206_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_3207_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K5 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K5 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3208_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_3209_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N2: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3210_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K5: nat] : ( times_times @ A @ ( gbinomial @ A @ A4 @ K5 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K5 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A4 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3211_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y8: B,Y: B,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R4 @ 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 @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R4 ) ) ) ) ).
% same_fstI
thf(fact_3212_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A13: set @ A,R1: set @ ( product_prod @ A @ A ),A23: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A4: B] :
( ( equiv_equiv @ A @ A13 @ R1 )
=> ( ( equiv_equiv @ B @ A23 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F2 )
=> ( ( member @ B @ A4 @ A23 )
=> ( 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_3213_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),A4: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A4 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_3214_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N4
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3215_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_3216_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_3217_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_3218_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_3219_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B > A,P3: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y3: B,X2: C] : ( F2 @ X2 @ Y3 )
@ ( product_swap @ C @ B @ P3 ) )
= ( product_case_prod @ C @ B @ A @ F2 @ P3 ) ) ).
% case_swap
thf(fact_3220_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_3221_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_3222_union__diff__assoc,axiom,
! [A: $tType,C6: multiset @ A,B5: multiset @ A,A3: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C6 @ B5 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) @ C6 )
= ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ) ).
% union_diff_assoc
thf(fact_3223_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F7 @ F8 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F8 @ F7 ) ) ) ) ).
% less_filter_def
thf(fact_3224_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F2: A > B > C,G: A > B > C,P3: product_prod @ A @ B] :
( ! [X3: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y2 )
= Q4 )
=> ( ( F2 @ X3 @ Y2 )
= ( G @ X3 @ Y2 ) ) )
=> ( ( P3 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P3 )
= ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_3225_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_3226_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X2: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [A8: B,B6: C] : ( F4 @ A8 @ B6 @ Y3 )
@ X2 ) ) ) ).
% nested_case_prod_simp
thf(fact_3227_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X13: A,X24: B] : ( H2 @ ( F2 @ X13 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_3228_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P3 )
= P3 ) ).
% case_prod_Pair_iden
thf(fact_3229_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X3: B,Y2: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X3 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_3230_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y3: B] : ( F2 @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_3231_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y2: B] :
( ( F2 @ X3 @ Y2 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y2 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_3232_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X13: A,X24: B] : X13 ) ) ).
% fst_def
thf(fact_3233_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > C] :
( ( ^ [X2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X2 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A8: A,Uu: B] : ( F2 @ A8 ) ) ) ).
% fn_fst_conv
thf(fact_3234_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C] :
( ( ^ [X2: product_prod @ A @ B] : ( F2 @ ( product_snd @ A @ B @ X2 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu: A] : F2 ) ) ).
% fn_snd_conv
thf(fact_3235_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X13: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_3236_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_3237_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P5: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P5 ) @ ( product_snd @ B @ C @ P5 ) ) ) ) ).
% case_prod_beta
thf(fact_3238_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_3239_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc5280177257484947105e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_3240_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X2: A,Y3: B] : ( product_Pair @ C @ D @ ( F4 @ X2 ) @ ( G4 @ Y3 ) ) ) ) ) ).
% map_prod_def
thf(fact_3241_congruentI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F2: A > B] :
( ! [Y2: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( ( F2 @ Y2 )
= ( F2 @ Z4 ) ) )
=> ( equiv_congruent @ A @ B @ R2 @ F2 ) ) ).
% congruentI
thf(fact_3242_congruentD,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F2: A > B,Y: A,Z2: A] :
( ( equiv_congruent @ A @ B @ R2 @ F2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( ( F2 @ Y )
= ( F2 @ Z2 ) ) ) ) ).
% congruentD
thf(fact_3243_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P3: product_prod @ B @ A,Q2: C > $o,F2: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P3 ) @ ( product_fst @ B @ A @ P3 ) )
=> ( ! [X3: B,Y2: A] :
( ( P @ Y2 @ X3 )
=> ( Q2 @ ( F2 @ X3 @ Y2 ) ) )
=> ( Q2 @ ( product_case_prod @ B @ A @ C @ F2 @ P3 ) ) ) ) ).
% exE_realizer
thf(fact_3244_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F2: A > B > C,G: D > A] :
( ( ^ [U3: product_prod @ D @ B] : ( F2 @ ( G @ ( product_fst @ D @ B @ U3 ) ) @ ( product_snd @ D @ B @ U3 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X2: D] : ( F2 @ ( G @ X2 ) ) ) ) ).
% split_comp_eq
thf(fact_3245_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X2: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_3246_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C5: A > B > C,P5: product_prod @ A @ B] : ( C5 @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% case_prod_unfold
thf(fact_3247_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 )
@ ^ [I4: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I4 ) )
@ A3 ) ).
% swap_inj_on
thf(fact_3248_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_3249_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_3250_swap__product,axiom,
! [B: $tType,A: $tType,A3: set @ B,B5: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I4: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I4 ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : A3 ) ) ).
% swap_product
thf(fact_3251_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F2: D > C > A,G: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F2 @ G ) @ X )
= ( F2 @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_3252_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G: D > B,A3: set @ C,B5: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X2: C,Y3: D] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G @ Y3 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu: C] : B5 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A3 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_3253_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,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_3254_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,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_3255_periodic__finite__ex,axiom,
! [D3: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K3: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D3 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_3256_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_3257_cpmi,axiom,
! [D4: int,P: int > $o,P7: int > $o,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z7: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z7 )
=> ( ( P @ X3 )
= ( P7 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P7 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ B5 )
& ( P @ ( plus_plus @ int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_3258_cppi,axiom,
! [D4: int,P: int > $o,P7: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z7: int] :
! [X3: int] :
( ( ord_less @ int @ Z7 @ X3 )
=> ( ( P @ X3 )
= ( P7 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P7 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ A3 )
& ( P @ ( minus_minus @ int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_3259_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_3260_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q6: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_3261_Sum__Icc__int,axiom,
! [M: int,N2: int] :
( ( ord_less_eq @ int @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M @ N2 ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N2 @ ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_3262_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),X4: set @ A,Y5: set @ A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F2 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X4 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ Y5 ) ) )
=> ( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 ) ) ) )
=> ( X4 = Y5 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_3263_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_3264_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_3265_Set__filter__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( filter3 @ A @ P @ A3 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X2: A,A17: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A17 ) @ A17 )
@ ( bot_bot @ ( set @ A ) )
@ A3 ) ) ) ).
% Set_filter_fold
thf(fact_3266_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X4: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S ) @ X4 )
= ( 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,A17: 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 ) @ A17 )
@ A17 ) )
@ X4
@ S ) ) ) ).
% insert_relcomp_union_fold
thf(fact_3267_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I: set @ A,F2: A > B,I2: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I )
& ( ( F2 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I2 @ I )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I @ ( insert2 @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I ) @ ( F2 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I @ ( insert2 @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3268_Sum__Ico__nat,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_3269_case__prodI2,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,C2: A > B > $o] :
( ! [A6: A,B2: B] :
( ( P3
= ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( C2 @ A6 @ B2 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P3 ) ) ).
% case_prodI2
thf(fact_3270_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_3271_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A6: A,B2: B] :
( ( P3
= ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( member @ C @ Z2 @ ( C2 @ A6 @ B2 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P3 ) ) ) ).
% mem_case_prodI2
thf(fact_3272_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A4: B,B3: C] :
( ( member @ A @ Z2 @ ( C2 @ A4 @ B3 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_3273_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A6: A,B2: B] :
( ( ( product_Pair @ A @ B @ A6 @ B2 )
= P3 )
=> ( C2 @ A6 @ B2 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ X ) ) ).
% case_prodI2'
thf(fact_3274_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q2: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B6: B] :
( ( P @ A8 )
& ( Q2 @ B6 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_3275_member__filter,axiom,
! [A: $tType,X: A,P: A > $o,A3: set @ A] :
( ( member @ A @ X @ ( filter3 @ A @ P @ A3 ) )
= ( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_3276_pair__set__inverse,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [B6: A,A8: B] : ( P @ A8 @ B6 ) ) ) ) ).
% pair_set_inverse
thf(fact_3277_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B6: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B6: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_3278_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_3279_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_3280_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_3281_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3282_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert2 @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3283_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I: set @ B,P3: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I )
& ( ( P3 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert2 @ B @ I2 @ I ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I ) ) )
& ( ~ ( member @ B @ I2 @ I )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert2 @ B @ I2 @ I ) )
= ( plus_plus @ A @ ( P3 @ I2 ) @ ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3284_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3285_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P2: A > $o,R3: 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 )
@ ^ [X11: A,Y9: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y3: B] :
( ( X11 = X2 )
& ( P2 @ X2 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y3 ) @ ( R3 @ X2 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_3286_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P3: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P3 ) )
=> ~ ! [X3: B,Y2: C] :
( ( P3
= ( product_Pair @ B @ C @ X3 @ Y2 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_3287_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 )
@ ^ [A8: A,B6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A18: A,B13: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A18 ) @ Ra )
| ( ( A8 = A18 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B6 @ B13 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_3288_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P2: A > $o,A5: set @ A] :
( collect @ A
@ ^ [A8: A] :
( ( member @ A @ A8 @ A5 )
& ( P2 @ A8 ) ) ) ) ) ).
% Set.filter_def
thf(fact_3289_mset__distrib,axiom,
! [A: $tType,A3: multiset @ A,B5: multiset @ A,M2: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A3 @ B5 )
= ( plus_plus @ ( multiset @ A ) @ M2 @ N ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A3
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B5
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M2
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_3290_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P3 )
=> ~ ! [X3: A,Y2: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
=> ~ ( C2 @ X3 @ Y2 ) ) ) ).
% case_prodE
thf(fact_3291_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_3292_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q2: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y3: B] :
( ( P @ X2 )
& ( Q2 @ X2 @ Y3 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X2: A] : ( collect @ B @ ( Q2 @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_3293_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P3: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ Z2 )
=> ~ ! [X3: A,Y2: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
=> ~ ( C2 @ X3 @ Y2 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_3294_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A4: A,B3: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A4 @ B3 ) @ C2 )
=> ( R4 @ A4 @ B3 @ C2 ) ) ).
% case_prodD'
thf(fact_3295_Id__on__def_H,axiom,
! [A: $tType,A3: A > $o] :
( ( id_on @ A @ ( collect @ A @ A3 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] :
( ( X2 = Y3 )
& ( A3 @ X2 ) ) ) ) ) ).
% Id_on_def'
thf(fact_3296_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
=> ( A3 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_3297_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_3298_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_3299_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3300_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,X2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% converse_unfold
thf(fact_3301_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V4: A,W3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V4 @ W3 ) @ R3 )
& ~ ( member @ A @ V4 @ A5 )
& ~ ( member @ A @ W3 @ A5 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_3302_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X2 ) @ ( F4 @ Y3 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_3303_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R5: 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 )
@ ^ [A15: A,A24: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B14: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A15 @ ( insert2 @ A @ A24 @ ( insert2 @ A @ B14 @ ( insert2 @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A15 = B14 )
& ( A24 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A15 @ A24 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A15 @ A24 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A15 @ B14 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A15 @ A24 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) )
& ( A15 = B14 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A24 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_3304_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_3305_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,P3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3306_wf__bounded__supset,axiom,
! [A: $tType,S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [Q8: set @ A,Q: set @ A] :
( ( ord_less @ ( set @ A ) @ Q @ Q8 )
& ( ord_less_eq @ ( set @ A ) @ Q8 @ S ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_3307_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_3308_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_3309_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X4: set @ A,A3: set @ ( product_prod @ A @ B ),Y5: set @ B,P: A > B > $o,Q2: A > B > $o] :
( ( X4
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
=> ( ( Y5
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y5 )
=> ( ( P @ X3 @ Xa3 )
=> ( Q2 @ X3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_3310_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3311_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A8: A,B6: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A8 @ B6 ) @ ( insert2 @ A @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3312_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3313_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: nat,B3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A4 @ ( suc @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A4 @ B3 ) ) @ ( G @ B3 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3314_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3315_atLeastLessThanSuc,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
= ( insert2 @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3316_rtrancl__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A4 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_3317_trancl__insert2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A4 ) @ ( transitive_trancl @ A @ R2 ) )
| ( X2 = A4 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
| ( Y3 = B3 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_3318_Image__fold,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R4 )
=> ( ( image @ A @ B @ R4 @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X2: A,Y3: B,A5: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S ) @ ( insert2 @ B @ Y3 @ A5 ) @ A5 ) )
@ ( bot_bot @ ( set @ B ) )
@ R4 ) ) ) ).
% Image_fold
thf(fact_3319_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E3: C > ( set @ A ),F5: D > ( set @ B ),A3: set @ C,B5: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: C,B6: D] :
( product_Sigma @ A @ B @ ( E3 @ A8 )
@ ^ [Uu: A] : ( F5 @ B6 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu: C] : B5 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E3 @ A3 ) )
@ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F5 @ B5 ) ) ) ) ).
% UN_Times_distrib
thf(fact_3320_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_3321_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3322_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_3323_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A8: A,B6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B6 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_3324_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3325_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_3326_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R4 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S )
=> ( ( relcomp @ A @ B @ C @ R4 @ 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,Y3: B,A5: 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,A17: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Z3 ) @ A17 ) @ A17 ) )
@ A5
@ S ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R4 ) ) ) ) ).
% relcomp_fold
thf(fact_3327_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R4: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R4 ) @ 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,A17: 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 ) @ A17 )
@ A17 ) )
@ ( relcomp @ C @ A @ B @ R4 @ S )
@ S ) ) ) ).
% insert_relcomp_fold
thf(fact_3328_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_3329_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I: set @ A,F2: A > B,I2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I )
& ( ( F2 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I2 @ I )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I @ ( insert2 @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I ) @ ( F2 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I @ ( insert2 @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3330_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
@ ^ [I4: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3331_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
@ ^ [I4: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3332_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X2: C,Y3: A,A5: 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,A17: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W3 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X2 @ Z3 ) @ A17 ) @ A17 ) )
@ A5
@ S ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_3333_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q6: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_3334_underS__Restr__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ A4 )
@ ^ [Uu: A] : ( order_underS @ A @ R2 @ A4 ) ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_3335_triangle__def,axiom,
( nat_triangle
= ( ^ [N4: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_3336_max__extp__max__ext__eq,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R4 ) )
= ( ^ [X2: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y3 ) @ ( max_ext @ A @ R4 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_3337_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B6: B] :
( P
& ( Q2 @ A8 @ B6 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_3338_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image2 @ code_integer @ code_integer
@ ^ [X2: code_integer] : ( plus_plus @ code_integer @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_3339_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_3340_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_3341_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_3342_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_3343_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X2: A,A17: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A17 ) @ A17 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_3344_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X2: A,Y3: B,A5: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S ) @ ( insert2 @ B @ Y3 @ A5 ) @ A5 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_3345_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R3: 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,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_3346_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X2: B,Z3: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) )
@ Z3
@ B5 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_3347_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K5: int] :
( if @ code_integer @ ( ord_less @ int @ K5 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K5 ) ) )
@ ( if @ code_integer
@ ( K5
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_3348_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y3: A] :
( ( ord_less @ nat @ ( F4 @ X2 ) @ ( F4 @ Y3 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X2 ) @ ( F4 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_3349_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S
@ ( insert2 @ A
@ ( ord_Least @ A
@ ^ [N4: A] : ( member @ A @ N4 @ S ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N2 ) ) ) ).
% enumerate_Suc
thf(fact_3350_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3351_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R4: A > B > $o,A4: D > A,B3: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z5: D] : Y4 = Z5
@ R4
@ A4
@ B3 )
= ( ? [Z3: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z3 )
= A4 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z3 )
= B3 ) ) ) ) ).
% fun.in_rel
thf(fact_3352_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B5: C > D > $o,F2: A > C,G: B > D] :
( ! [X3: A,Y2: B] :
( ( A3 @ X3 @ Y2 )
=> ( B5 @ ( F2 @ X3 ) @ ( G @ Y2 ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B5 @ F2 @ G ) ) ).
% rel_funI
thf(fact_3353_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% finite_atLeastLessThan_integer
thf(fact_3354_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% finite_atLeastAtMost_integer
thf(fact_3355_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R4: A > B > $o] :
( ( R4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R4 @ ( bNF_rel_fun @ A @ B @ A @ B @ R4 @ R4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z5: num] : Y4 = Z5
@ R4
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_3356_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).
% finite_atLeastZeroLessThan_integer
thf(fact_3357_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R4: A > B > $o] :
( ( R4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R4 @ ( bNF_rel_fun @ A @ B @ A @ B @ R4 @ R4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R4
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ R4 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_3358_times__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ ^ [Y4: int,Z5: int] : Y4 = Z5 )
@ ( times_times @ int )
@ ( times_times @ int ) ) ).
% times_integer.rsp
thf(fact_3359_times__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5 )
@ ( times_times @ nat )
@ ( times_times @ nat ) ) ).
% times_natural.rsp
thf(fact_3360_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_3361_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q2: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_3362_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_3363_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A,Q2: A > $o] :
( ( P @ A4 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_3364_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A3: C > D > $o,B5: C > D > $o,F2: A > C,G: B > D] :
( ! [X3: A,Y2: B] :
( ( Y5 @ X3 @ Y2 )
=> ( X4 @ X3 @ Y2 ) )
=> ( ! [X3: C,Y2: D] :
( ( A3 @ X3 @ Y2 )
=> ( B5 @ X3 @ Y2 ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A3 @ F2 @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B5 @ F2 @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_3365_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A3: C > D > $o,F2: A > C,G: B > D,Y5: A > B > $o,B5: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A3 @ F2 @ G )
=> ( ! [X3: A,Y2: B] :
( ( Y5 @ X3 @ Y2 )
=> ( X4 @ X3 @ Y2 ) )
=> ( ! [X3: C,Y2: D] :
( ( A3 @ X3 @ Y2 )
=> ( B5 @ X3 @ Y2 ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B5 @ F2 @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_3366_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B5: C > D > $o,F2: A > C,G: B > D,X: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B5 @ F2 @ G )
=> ( ( A3 @ X @ Y )
=> ( B5 @ ( F2 @ X ) @ ( G @ Y ) ) ) ) ).
% rel_funD
thf(fact_3367_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R4: A > B > $o] :
( ( R4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R4 @ ( bNF_rel_fun @ A @ B @ A @ B @ R4 @ R4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R4 @ R4 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y4: int,Z5: int] : Y4 = Z5
@ R4
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_3368_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q2: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B15: A] :
( ( P @ B15 )
=> ( ord_less_eq @ A @ A6 @ B15 ) )
=> ( Q2 @ A6 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_3369_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A,Q2: A > $o] :
( ( P @ A4 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B15: A] :
( ( P @ B15 )
=> ( ord_less_eq @ A @ A6 @ B15 ) )
=> ( Q2 @ A6 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_3370_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_3371_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ X3 @ Y6 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_3372_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X6 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya: A] :
( ( P @ Ya )
=> ( ord_less_eq @ A @ Y2 @ Ya ) ) )
=> ( Y2 = X6 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_3373_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X6 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya: A] :
( ( P @ Ya )
=> ( ord_less_eq @ A @ Y2 @ Ya ) ) )
=> ( Y2 = X6 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_3374_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_3375_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_3376_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R12: A > B > $o,R23: C > D > $o,P: A > C > $o,Q2: B > D > $o,A4: product_prod @ A @ B,A3: set @ ( product_prod @ A @ B ),B3: product_prod @ C @ D,B5: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R12
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R23
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ P
@ Q2 )
=> ( ( member @ ( product_prod @ A @ B ) @ A4 @ A3 )
=> ( ( member @ ( product_prod @ C @ D ) @ B3 @ B5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R12 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B5 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R23 ) ) )
=> ( ( P @ ( product_fst @ A @ B @ A4 ) @ ( product_fst @ C @ D @ B3 ) )
= ( Q2 @ ( product_snd @ A @ B @ A4 ) @ ( product_snd @ C @ D @ B3 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_3377_times__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_3378_mlex__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R4: 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 @ R4 ) ) ) ).
% mlex_less
thf(fact_3379_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R4 ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 ) ) ) ) ).
% mlex_iff
thf(fact_3380_mlex__leq,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R4 ) ) ) ) ).
% mlex_leq
thf(fact_3381_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A3: A > B > $o,B5: C > D > $o,F2: A > C,G: B > D,X4: set @ ( product_prod @ A @ B ),X: product_prod @ A @ B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B5 @ F2 @ G )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
=> ( ( member @ ( product_prod @ A @ B ) @ X @ X4 )
=> ( B5 @ ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) @ X ) @ ( comp @ B @ D @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) @ X ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_3382_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A5: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A5 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A5 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_3383_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R4: A > B > $o] :
( ( R4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R4 @ ( bNF_rel_fun @ A @ B @ A @ B @ R4 @ R4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ R4
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_3384_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R4: A > B > $o] :
( ( R4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5
@ R4
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_3385_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,T3: A > B > $o] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( T3
= ( ^ [X2: A,Y3: B] :
( X2
= ( Rep2 @ Y3 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T3
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ ^ [X2: A] : X2
@ Rep2 ) ) ) ).
% typedef_rep_transfer
thf(fact_3386_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K5: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K5
= ( 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 ) @ K5 )
@ ( 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 @ K5 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K5 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S7 ) ) )
@ ( code_divmod_abs @ K5 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_3387_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,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_3388_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,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_3389_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K5: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K5 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K5 ) ) )
@ ( if @ int
@ ( K5
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K5 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_3390_positive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ positive ) ).
% positive.transfer
thf(fact_3391_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,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y3 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V4 ) @ ( times_times @ nat @ Y3 @ U3 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_3392_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_3393_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,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y3 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V4 ) @ ( times_times @ nat @ Y3 @ U3 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_3394_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa ) )
= ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% times_integer.rep_eq
thf(fact_3395_positive__mult,axiom,
! [X: rat,Y: rat] :
( ( positive @ X )
=> ( ( positive @ Y )
=> ( positive @ ( times_times @ rat @ X @ Y ) ) ) ) ).
% positive_mult
thf(fact_3396_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_3397_type__definition__integer,axiom,
type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% type_definition_integer
thf(fact_3398_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_3399_positive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% positive.rep_eq
thf(fact_3400_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_3401_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_3402_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% diff_numeral_special(8)
thf(fact_3403_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( neg_numeral_sub @ A @ N2 @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_3404_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X11: A,Y9: B] :
( ( X = X11 )
& ( Y = Y9 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_3405_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( neg_numeral_sub @ A @ one2 @ N2 ) ) ) ).
% diff_numeral_special(1)
thf(fact_3406_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_3407_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( neg_numeral_sub @ A @ N2 @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_3408_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_3409_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_3410_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_3411_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_3412_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_3413_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_3414_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_prod_def
thf(fact_3415_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_3416_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_3417_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_3418_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I: set @ B,P3: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I )
& ( ( P3 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert2 @ B @ I2 @ I ) )
= ( groups1962203154675924110t_prod @ B @ A @ P3 @ I ) ) )
& ( ~ ( member @ B @ I2 @ I )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert2 @ B @ I2 @ I ) )
= ( times_times @ A @ ( P3 @ I2 ) @ ( groups1962203154675924110t_prod @ B @ A @ P3 @ I ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_3419_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_3420_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_3421_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,I: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I ) ) ) ).
% prod.non_neutral'
thf(fact_3422_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_3423_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_3424_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_3425_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T3 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_3426_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T3: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( H2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_3427_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T3: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T3 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_3428_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I )
& ( ( H2 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_3429_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P5: B > A,I5: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P5
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_3430_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_3431_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_3432_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3433_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,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y3 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V4 ) @ ( times_times @ nat @ Y3 @ U3 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y3 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V4 ) @ ( times_times @ nat @ Y3 @ U3 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_3434_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X4: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ X4 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X2: A] : ( principal @ B @ ( F2 @ X2 ) )
@ X4 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X4 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_3435_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_3436_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3437_principal__inject,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( principal @ A @ A3 )
= ( principal @ A @ B5 ) )
= ( A3 = B5 ) ) ).
% principal_inject
thf(fact_3438_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M: A,N2: A] :
( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N2 ) )
= ( ( ord_min @ A @ M @ N2 )
= M ) ) ) ).
% min_arg_le(2)
thf(fact_3439_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N2: A,M: A] :
( ( ord_less_eq @ A @ N2 @ ( ord_min @ A @ M @ N2 ) )
= ( ( ord_min @ A @ M @ N2 )
= N2 ) ) ) ).
% min_arg_le(1)
thf(fact_3440_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ( ord_min @ A @ M @ N2 )
= N2 )
= ( ord_less_eq @ A @ N2 @ M ) ) ) ).
% min_eq_arg(2)
thf(fact_3441_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ( ord_min @ A @ M @ N2 )
= M )
= ( ord_less_eq @ A @ M @ N2 ) ) ) ).
% min_eq_arg(1)
thf(fact_3442_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ).
% min_simps(2)
thf(fact_3443_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ).
% min_simps(1)
thf(fact_3444_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ B3 )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% min_less_self_conv(2)
thf(fact_3445_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ A4 )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% min_less_self_conv(1)
thf(fact_3446_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N2 ) @ N2 ) )
= ( ( ord_min @ A @ M @ N2 )
= N2 ) ) ) ).
% min_arg_not_ge(2)
thf(fact_3447_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N2 ) @ M ) )
= ( ( ord_min @ A @ M @ N2 )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_3448_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_3449_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_3450_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_3451_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_3452_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_3453_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_3454_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_3455_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_3456_sup__principal,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B5 ) )
= ( principal @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% sup_principal
thf(fact_3457_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_3458_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_3459_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_3460_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_3461_min__Suc__gt_I1_J,axiom,
! [A4: nat,B3: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_min @ nat @ ( suc @ A4 ) @ B3 )
= ( suc @ A4 ) ) ) ).
% min_Suc_gt(1)
thf(fact_3462_min__Suc__gt_I2_J,axiom,
! [A4: nat,B3: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_min @ nat @ B3 @ ( suc @ A4 ) )
= ( suc @ A4 ) ) ) ).
% min_Suc_gt(2)
thf(fact_3463_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3464_principal__le__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% principal_le_iff
thf(fact_3465_inf__principal,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B5 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% inf_principal
thf(fact_3466_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_3467_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3468_SUP__principal,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I4: B] : ( principal @ A @ ( A3 @ I4 ) )
@ I ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A3 @ I ) ) ) ) ).
% SUP_principal
thf(fact_3469_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_3470_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_3471_nat__mult__min__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N2 @ Q4 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_3472_nat__mult__min__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N2 ) @ Q4 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_3473_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A8: A,B6: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B6 ) @ A8 @ B6 ) ) ) ) ).
% min_def
thf(fact_3474_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_3475_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_3476_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_3477_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_3478_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_3479_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_3480_top__eq__principal__UNIV,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( principal @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% top_eq_principal_UNIV
thf(fact_3481_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_3482_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_3483_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan @ nat @ N2 )
= ( bot_bot @ ( set @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3484_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N2: A] :
( ( ( set_ord_lessThan @ A @ N2 )
= ( bot_bot @ ( set @ A ) ) )
= ( N2
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_3485_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_3486_principal__eq__bot__iff,axiom,
! [A: $tType,X4: set @ A] :
( ( ( principal @ A @ X4 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_3487_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_3488_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_3489_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_3490_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_3491_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_3492_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_3493_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_3494_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N: set @ A] :
( ! [X3: A,Y2: A] :
( ( H2 @ ( ord_min @ A @ X3 @ Y2 ) )
= ( ord_min @ A @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H2 @ N ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_3495_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset
thf(fact_3496_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_3497_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Min.closed
thf(fact_3498_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( lattic643756798350308766er_Min @ A @ B5 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_3499_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A3 ) ) ) ) ).
% Min.eq_fold
thf(fact_3500_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3501_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M4 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_3502_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M4 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_3503_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_3504_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_3505_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_3506_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_3507_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
=> ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
= ( image2 @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_integer
thf(fact_3508_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( 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 @ N2 ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3509_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3510_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N2: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3511_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3512_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3513_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3514_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N2 ) ) @ ( power_power @ A @ Y @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X @ P5 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3515_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K5: A] : ( principal @ A @ ( set_ord_atMost @ A @ K5 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_3516_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K5: A] : ( principal @ A @ ( set_ord_atMost @ A @ K5 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_3517_finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ 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_3518_bezw__aux,axiom,
! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_3519_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_3520_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_3521_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_3522_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_3523_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( gcd_gcd @ A @ A4 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ).
% Gcd_insert
thf(fact_3524_Gcd__fin_Oinsert,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 @ A3 ) ) ) ) ).
% Gcd_fin.insert
thf(fact_3525_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_3526_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_3527_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N2 ) )
= ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_3528_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_3529_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K: A,N2: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N2 ) )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_add_mult
thf(fact_3530_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_3531_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_3532_Gcd__in,axiom,
! [A3: set @ nat] :
( ! [A6: nat,B2: nat] :
( ( member @ nat @ A6 @ A3 )
=> ( ( member @ nat @ B2 @ A3 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A6 @ B2 ) @ A3 ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ).
% Gcd_in
thf(fact_3533_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 )
= ( gcd_gcd @ A @ B3 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_3534_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B3 @ ( times_times @ A @ C2 @ A4 ) )
= ( gcd_gcd @ A @ B3 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_3535_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B3 @ ( divide_divide @ A @ C2 @ A4 ) )
= ( gcd_gcd @ A @ B3 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_3536_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B3 @ A4 ) @ C2 )
= ( gcd_gcd @ A @ B3 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_3537_bezout__nat,axiom,
! [A4: nat,B3: nat] :
( ( A4
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y2: nat] :
( ( times_times @ nat @ A4 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ ( gcd_gcd @ nat @ A4 @ B3 ) ) ) ) ).
% bezout_nat
thf(fact_3538_bezout__gcd__nat_H,axiom,
! [B3: nat,A4: nat] :
? [X3: nat,Y2: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B3 @ Y2 ) @ ( times_times @ nat @ A4 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A4 @ X3 ) @ ( times_times @ nat @ B3 @ Y2 ) )
= ( gcd_gcd @ nat @ A4 @ B3 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A4 @ Y2 ) @ ( times_times @ nat @ B3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B3 @ X3 ) @ ( times_times @ nat @ A4 @ Y2 ) )
= ( gcd_gcd @ nat @ A4 @ B3 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_3539_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_3540_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_3541_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_3542_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A5: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X7: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y10: set @ A] :
( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X7 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A5 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
& ( ord_less_eq @ ( set @ A ) @ X7 @ A5 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_3543_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A5: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A5 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_3544_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_3545_wf__finite__segments,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) )
=> ( wf @ A @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_3546_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,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X2: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% plus_rat.rsp
thf(fact_3547_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P2: A > A > $o,A5: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A8: A,B6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
& ( P2 @ A8 @ B6 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_3548_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I: set @ A,F5: A > ( set @ B ),F2: B > C,G5: D > ( set @ C ),J: set @ D] :
( ( I
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I )
=> ! [J4: A] :
( ( member @ A @ J4 @ I )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I3 ) @ ( F5 @ J4 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J4 ) @ ( F5 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F2
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G5 @ J3 ) )
@ J ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F5 @ I4 ) )
@ I ) ) )
= ( ! [X2: D] :
( ( member @ D @ X2 @ J )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ I )
& ! [Z3: B] :
( ( member @ B @ Z3 @ ( F5 @ Y3 ) )
=> ( member @ C @ ( F2 @ Z3 ) @ ( G5 @ X2 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_3549_positive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% positive.rsp
thf(fact_3550_ratrel__iff,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y3: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y3 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_3551_filterlim__ident,axiom,
! [A: $tType,F5: filter @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ F5
@ F5 ) ).
% filterlim_ident
thf(fact_3552_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F32: filter @ B,F23: filter @ A,F2: C > A,F13: filter @ C] :
( ( filterlim @ A @ B @ G @ F32 @ F23 )
=> ( ( filterlim @ C @ A @ F2 @ F23 @ F13 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ F32
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_3553_filterlim__mono,axiom,
! [B: $tType,A: $tType,F2: A > B,F23: filter @ B,F13: filter @ A,F24: filter @ B,F14: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F23 @ F13 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F23 @ F24 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F14 @ F13 )
=> ( filterlim @ A @ B @ F2 @ F24 @ F14 ) ) ) ) ).
% filterlim_mono
thf(fact_3554_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A8: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A8 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_3555_irreflI,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R4 )
=> ( irrefl @ A @ R4 ) ) ).
% irreflI
thf(fact_3556_filterlim__sup,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ B,F13: filter @ A,F23: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F5 @ F13 )
=> ( ( filterlim @ A @ B @ F2 @ F5 @ F23 )
=> ( filterlim @ A @ B @ F2 @ F5 @ ( sup_sup @ ( filter @ A ) @ F13 @ F23 ) ) ) ) ).
% filterlim_sup
thf(fact_3557_filterlim__top,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ A] : ( filterlim @ A @ B @ F2 @ ( top_top @ ( filter @ B ) ) @ F5 ) ).
% filterlim_top
thf(fact_3558_filterlim__inf,axiom,
! [B: $tType,A: $tType,F2: A > B,F23: filter @ B,F32: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( inf_inf @ ( filter @ B ) @ F23 @ F32 ) @ F13 )
= ( ( filterlim @ A @ B @ F2 @ F23 @ F13 )
& ( filterlim @ A @ B @ F2 @ F32 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_3559_bezout__int,axiom,
! [X: int,Y: int] :
? [U4: int,V3: int] :
( ( plus_plus @ int @ ( times_times @ int @ U4 @ X ) @ ( times_times @ int @ V3 @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% bezout_int
thf(fact_3560_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N2: int] :
( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N2 ) )
= ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) ) ).
% gcd_mult_distrib_int
thf(fact_3561_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G5: C > ( filter @ B ),B5: set @ C,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G5 @ B5 ) ) @ F5 )
= ( ! [X2: C] :
( ( member @ C @ X2 @ B5 )
=> ( filterlim @ A @ B @ F2 @ ( G5 @ X2 ) @ F5 ) ) ) ) ).
% filterlim_INF
thf(fact_3562_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A3: set @ A,F2: B > C,F5: filter @ C,G5: A > ( filter @ B )] :
( ( member @ A @ X @ A3 )
=> ( ( filterlim @ B @ C @ F2 @ F5 @ ( G5 @ X ) )
=> ( filterlim @ B @ C @ F2 @ F5 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G5 @ A3 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_3563_filterlim__If,axiom,
! [B: $tType,A: $tType,F2: A > B,G5: filter @ B,F5: filter @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F2 @ G5 @ ( inf_inf @ ( filter @ A ) @ F5 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G5
@ ( inf_inf @ ( filter @ A ) @ F5
@ ( principal @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G5
@ F5 ) ) ) ).
% filterlim_If
thf(fact_3564_irrefl__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( irrefl @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ).
% irrefl_diff_Id
thf(fact_3565_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J: set @ A,I2: A > C,I: set @ C,F5: C > ( set @ D ),F2: D > E,G5: A > ( set @ E )] :
( ! [M3: A,X3: B] :
( ( member @ A @ M3 @ J )
=> ( member @ C @ ( I2 @ M3 ) @ I ) )
=> ( ! [M3: A,X3: D] :
( ( member @ A @ M3 @ J )
=> ( ( member @ D @ X3 @ ( F5 @ ( I2 @ M3 ) ) )
=> ( member @ E @ ( F2 @ X3 ) @ ( G5 @ M3 ) ) ) )
=> ( filterlim @ D @ E @ F2
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J3: A] : ( principal @ E @ ( G5 @ J3 ) )
@ J ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I4: C] : ( principal @ D @ ( F5 @ I4 ) )
@ I ) ) ) ) ) ).
% filterlim_base
thf(fact_3566_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y3: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y3 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_3567_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,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X2: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% times_rat.rsp
thf(fact_3568_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_3569_positive_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% positive.abs_eq
thf(fact_3570_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_3571_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] :
? [A8: B,B6: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F2 @ A8 ) @ ( F2 @ B6 ) ) )
& ( P @ A8 @ B6 ) ) ) ) ).
% pairself_image_eq
thf(fact_3572_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B6: A,A5: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6
@ ( the @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_3573_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: B,B5: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [L2: B] :
( ( Uu
= ( F2 @ L2 ) )
& ( ( L2 = A4 )
| ( member @ B @ L2 @ B5 ) ) ) )
= ( insert2 @ A @ ( F2 @ A4 )
@ ( collect @ A
@ ^ [Uu: A] :
? [L2: B] :
( ( Uu
= ( F2 @ L2 ) )
& ( member @ B @ L2 @ B5 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_3574_ex__assn__proper,axiom,
! [A: $tType,P: A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [X3: A] : ( proper @ ( P @ X3 ) )
=> ( proper
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X2: A] : ( P @ X2 @ H4 ) ) ) ).
% ex_assn_proper
thf(fact_3575_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
= ( collect @ A
@ ^ [X2: A] :
? [X7: B] : ( P @ X2 @ X7 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_3576_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( range2 @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: B] : ( P @ X2 @ Y3 ) ) ) ).
% Range_Collect_case_prod
thf(fact_3577_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X2: B] :
( ( Uu
= ( F2 @ X2 ) )
& ( member @ B @ X2 @ A3 ) ) )
= ( image2 @ B @ A @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_3578_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X2: B] :
( ( Uu
= ( F2 @ X2 ) )
& ( P @ X2 ) ) )
= ( image2 @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_3579_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B > C,S: set @ C] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [P5: product_prod @ A @ B] :
( ( Uu = P5 )
& ( member @ C @ ( F2 @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) @ S ) ) ) )
= ( collect @ A
@ ^ [A8: A] :
? [B6: B] : ( member @ C @ ( F2 @ A8 @ B6 ) @ S ) ) ) ).
% fs_contract
thf(fact_3580_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( collect @ A
@ ^ [U3: A] :
? [X2: B] :
( U3
= ( F2 @ X2 ) ) )
= ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_3581_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P5: product_prod @ A @ A] :
? [X2: A] :
( P5
= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_def
thf(fact_3582_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X2: A] :
? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ).
% Domain_unfold
thf(fact_3583_Pow__Compl,axiom,
! [A: $tType,A3: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [B8: set @ A] :
( ( Uu
= ( uminus_uminus @ ( set @ A ) @ B8 ) )
& ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B8 ) ) ) ) ) ).
% Pow_Compl
thf(fact_3584_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B1: A,B22: A,B32: A,F2: A > B] :
( ( ord_less_eq @ A @ B1 @ B22 )
=> ( ( ord_less_eq @ A @ B22 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [I4: A] :
( ( Uu
= ( F2 @ I4 ) )
& ( ord_less_eq @ A @ B1 @ I4 )
& ( ord_less @ A @ I4 @ B22 ) ) )
@ ( collect @ B
@ ^ [Uu: B] :
? [I4: A] :
( ( Uu
= ( F2 @ I4 ) )
& ( ord_less_eq @ A @ B22 @ I4 )
& ( ord_less @ A @ I4 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu: B] :
? [I4: A] :
( ( Uu
= ( F2 @ I4 ) )
& ( ord_less_eq @ A @ B1 @ I4 )
& ( ord_less @ A @ I4 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_3585_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S7: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Z3: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y3 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z3 ) @ S7 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_3586_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A8: A,B6: A] :
( ( Uu
= ( inf_inf @ A @ A8 @ B6 ) )
& ( member @ A @ A8 @ A3 )
& ( member @ A @ B6 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_3587_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A8: A] :
( ( Uu
= ( inf_inf @ A @ X @ A8 ) )
& ( member @ A @ A8 @ A3 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_3588_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A8: A,B6: A] :
( ( Uu
= ( sup_sup @ A @ A8 @ B6 ) )
& ( member @ A @ A8 @ A3 )
& ( member @ A @ B6 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_3589_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A8: A] :
( ( Uu
= ( sup_sup @ A @ X @ A8 ) )
& ( member @ A @ A8 @ A3 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_3590_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R3: 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,Y3: B] :
( ( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X2 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R3 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X2: A,Y3: B] :
( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X2 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_3591_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A5: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A8: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F4 @ A8 ) @ ( G4 @ A8 ) ) )
& ( member @ C @ A8 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_3592_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,G: C > B,A3: set @ A,B5: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A8: A,B6: C] : ( times_times @ B @ ( F2 @ A8 ) @ ( G @ B6 ) ) )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu: A] : B5 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B5 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [A8: A,B6: C] :
( ( Uu
= ( times_times @ B @ ( F2 @ A8 ) @ ( G @ B6 ) ) )
& ( member @ A @ A8 @ A3 )
& ( member @ C @ B6 @ B5 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_3593_ex__assn__def,axiom,
! [A: $tType] :
( ( ex_assn @ A )
= ( ^ [P2: A > assn] :
( abs_assn
@ ^ [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X2: A] : ( rep_assn @ ( P2 @ X2 ) @ H4 ) ) ) ) ).
% ex_assn_def
thf(fact_3594_mset__set__Union,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A3 ) @ ( mset_set @ A @ B5 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_3595_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_3596_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% finite_greaterThanLessThan_integer
thf(fact_3597_ex__assn__const,axiom,
! [A: $tType,C2: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : C2 )
= C2 ) ).
% ex_assn_const
thf(fact_3598_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_3599_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_3600_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_3601_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_3602_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_3603_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_3604_mset__set_Oempty,axiom,
! [A: $tType] :
( ( mset_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset_set.empty
thf(fact_3605_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_3606_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_3607_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F4: A > C] : ( product_map_prod @ A @ C @ B @ B @ F4 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_3608_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_3609_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V: A] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A] :
( ( rep_assn @ ( P @ X3 ) @ H )
=> ( X3 = V ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V ) ) ) ).
% ex_one_point_gen
thf(fact_3610_ex__distrib__star,axiom,
! [A: $tType,P: A > assn,Q2: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( times_times @ assn @ ( P @ X2 ) @ Q2 ) )
= ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q2 ) ) ).
% ex_distrib_star
thf(fact_3611_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep2: A > B,Abs2: B > A] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ A @ A @ Abs2 @ Rep2 )
= ( id @ A ) ) ) ).
% type_copy_Abs_o_Rep
thf(fact_3612_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ A @ B @ B @ Rep2 @ Abs2 )
= ( id @ B ) ) ) ).
% type_copy_Rep_o_Abs
thf(fact_3613_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep2: A > B,Abs2: B > A,M2: B > B] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( M2
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs2 @ M2 ) @ Rep2 )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_3614_ex__distrib__and,axiom,
! [A: $tType,P: A > assn,Q2: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( inf_inf @ assn @ ( P @ X2 ) @ Q2 ) )
= ( inf_inf @ assn @ ( ex_assn @ A @ P ) @ Q2 ) ) ).
% ex_distrib_and
thf(fact_3615_ex__distrib__or,axiom,
! [A: $tType,P: A > assn,Q2: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( sup_sup @ assn @ ( P @ X2 ) @ Q2 ) )
= ( sup_sup @ assn @ ( ex_assn @ A @ P ) @ Q2 ) ) ).
% ex_distrib_or
thf(fact_3616_ex__join__or,axiom,
! [A: $tType,P: A > assn,Q2: A > assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( sup_sup @ assn @ ( P @ X2 ) @ ( ex_assn @ A @ Q2 ) ) )
= ( ex_assn @ A
@ ^ [X2: A] : ( sup_sup @ assn @ ( P @ X2 ) @ ( Q2 @ X2 ) ) ) ) ).
% ex_join_or
thf(fact_3617_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_3618_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_3619_mset__set__empty__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( mset_set @ A @ A3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% mset_set_empty_iff
thf(fact_3620_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_3621_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_3622_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_3623_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_3624_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F2: B > A,X: B,C2: C,G: B > C,A3: set @ B] :
( ( B3
= ( F2 @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F2 @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_3625_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_3626_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_3627_positive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% positive_def
thf(fact_3628_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A5: set @ A,A8: A] :
( ( X2
= ( insert2 @ A @ A8 @ A5 ) )
& ( P5 @ A5 ) ) ) ) ) ).
% finite_def
thf(fact_3629_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I2: A,C6: set @ A,G: A > B,X: A] :
( ( member @ A @ I2 @ C6 )
=> ( ( inj_on @ A @ B @ G @ C6 )
=> ( ( comp @ B @ A @ A
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G @ C6 ) ) @ ( the_inv_into @ A @ B @ C6 @ G @ I4 ) @ X )
@ G
@ I2 )
= ( id @ A @ I2 ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3630_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F2: A > B,Z2: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 @ Z2 )
= ( id @ A @ Z2 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_3631_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C6: set @ A,B5: set @ A,X: A] :
( ( inj_on @ A @ B @ G @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ C6 @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G @ C6 ) ) @ ( the_inv_into @ A @ B @ C6 @ G @ I4 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3632_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B5: set @ A,F5: A > ( filter @ B ),P: B > $o] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ B5 )
=> ! [B2: A] :
( ( member @ A @ B2 @ B5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ B5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X6 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A6 ) @ ( F5 @ B2 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ B5 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ B5 )
& ( eventually @ B @ P @ ( F5 @ X2 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_3633_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_3634_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_3635_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% finite_greaterThanAtMost_integer
thf(fact_3636_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X7: A] : ( P @ X7 ) ) ) ).
% eventually_top
thf(fact_3637_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_3638_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_3639_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_3640_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_3641_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_3642_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 )
=> ( P @ X8 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_3643_eventually__principal,axiom,
! [A: $tType,P: A > $o,S: set @ A] :
( ( eventually @ A @ P @ ( principal @ A @ S ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% eventually_principal
thf(fact_3644_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C6
=> ( P @ X2 ) )
@ F5 )
= ( C6
=> ( eventually @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_3645_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C6
| ( P @ X2 ) )
@ F5 )
= ( C6
| ( eventually @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_3646_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
| C6 )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
| C6 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_3647_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
@ F5 )
=> ( ( eventually @ A @ P @ F5 )
=> ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_mp
thf(fact_3648_eventually__True,axiom,
! [A: $tType,F5: filter @ A] :
( eventually @ A
@ ^ [X2: A] : $true
@ F5 ) ).
% eventually_True
thf(fact_3649_eventually__conj,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A @ Q2 @ F5 )
=> ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q2 @ X2 ) )
@ F5 ) ) ) ).
% eventually_conj
thf(fact_3650_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o,R4: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A @ Q2 @ F5 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( ( Q2 @ I3 )
=> ( R4 @ I3 ) ) )
=> ( eventually @ A @ R4 @ F5 ) ) ) ) ).
% eventually_elim2
thf(fact_3651_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [N4: A] :
( ( P @ N4 )
= ( Q2 @ N4 ) )
@ F5 )
=> ( ( eventually @ A @ P @ F5 )
= ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_subst
thf(fact_3652_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
@ F5 )
=> ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_rev_mp
thf(fact_3653_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q2 @ X2 ) )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
& ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_conj_iff
thf(fact_3654_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ~ ( eventually @ A @ Q2 @ F5 )
=> ~ ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
@ F5 ) ) ) ).
% not_eventually_impI
thf(fact_3655_eventuallyI,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F5 ) ) ).
% eventuallyI
thf(fact_3656_filter__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: filter @ A,Z5: filter @ A] : Y4 = Z5 )
= ( ^ [F7: filter @ A,F8: filter @ A] :
! [P2: A > $o] :
( ( eventually @ A @ P2 @ F7 )
= ( eventually @ A @ P2 @ F8 ) ) ) ) ).
% filter_eq_iff
thf(fact_3657_eventually__mono,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( eventually @ A @ Q2 @ F5 ) ) ) ).
% eventually_mono
thf(fact_3658_not__eventuallyD,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ~ ( eventually @ A @ P @ F5 )
=> ? [X3: A] :
~ ( P @ X3 ) ) ).
% not_eventuallyD
thf(fact_3659_always__eventually,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F5 ) ) ).
% always_eventually
thf(fact_3660_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
! [P2: A > $o] :
( ( eventually @ A @ P2 @ F8 )
=> ( eventually @ A @ P2 @ F7 ) ) ) ) ).
% le_filter_def
thf(fact_3661_filter__leI,axiom,
! [A: $tType,F9: filter @ A,F5: filter @ A] :
( ! [P4: A > $o] :
( ( eventually @ A @ P4 @ F9 )
=> ( eventually @ A @ P4 @ F5 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F5 @ F9 ) ) ).
% filter_leI
thf(fact_3662_filter__leD,axiom,
! [A: $tType,F5: filter @ A,F9: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F9 )
=> ( ( eventually @ A @ P @ F9 )
=> ( eventually @ A @ P @ F5 ) ) ) ).
% filter_leD
thf(fact_3663_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_3664_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_3665_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_3666_eventually__inf,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,F9: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F5 @ F9 ) )
= ( ? [Q: A > $o,R3: A > $o] :
( ( eventually @ A @ Q @ F5 )
& ( eventually @ A @ R3 @ F9 )
& ! [X2: A] :
( ( ( Q @ X2 )
& ( R3 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_inf
thf(fact_3667_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
? [X7: B] : ( P @ X2 @ X7 )
@ F5 )
= ( ? [Y10: A > B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ ( Y10 @ X2 ) )
@ F5 ) ) ) ).
% eventually_ex
thf(fact_3668_eventually__sup,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,F9: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F5 @ F9 ) )
= ( ( eventually @ A @ P @ F5 )
& ( eventually @ A @ P @ F9 ) ) ) ).
% eventually_sup
thf(fact_3669_eventually__Sup,axiom,
! [A: $tType,P: A > $o,S: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Sup_Sup @ ( filter @ A ) @ S ) )
= ( ! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S )
=> ( eventually @ A @ P @ X2 ) ) ) ) ).
% eventually_Sup
thf(fact_3670_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F25: filter @ B,F15: filter @ A] :
! [P2: B > $o] :
( ( eventually @ B @ P2 @ F25 )
=> ( eventually @ A
@ ^ [X2: A] : ( P2 @ ( F4 @ X2 ) )
@ F15 ) ) ) ) ).
% filterlim_iff
thf(fact_3671_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F14: filter @ A,F23: filter @ B,F24: filter @ B,F2: B > A,G: B > A] :
( ( F13 = F14 )
=> ( ( F23 = F24 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F23 )
=> ( ( filterlim @ B @ A @ F2 @ F13 @ F23 )
= ( filterlim @ B @ A @ G @ F14 @ F24 ) ) ) ) ) ).
% filterlim_cong
thf(fact_3672_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: filter @ A,F2: B > A,G5: filter @ B] :
( ( eventually @ A @ P @ F5 )
=> ( ( filterlim @ B @ A @ F2 @ F5 @ G5 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F2 @ X2 ) )
@ G5 ) ) ) ).
% eventually_compose_filterlim
thf(fact_3673_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_3674_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_3675_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_3676_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_3677_Func__is__emp,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A3 @ B5 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_3678_Func__non__emp,axiom,
! [A: $tType,B: $tType,B5: set @ A,A3: set @ B] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A3 @ B5 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_3679_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N4 @ N7 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_3680_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N4: A] :
( ( ord_less @ A @ N4 @ N7 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_3681_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( P @ A3 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_3682_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_3683_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_3684_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_3685_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_3686_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_3687_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ B,G5: filter @ A,F9: filter @ B,G6: filter @ A,F10: A > B] :
( ( filterlim @ A @ B @ F2 @ F5 @ G5 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F5 @ F9 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G6 @ G5 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( F10 @ X2 ) )
@ G6 )
=> ( filterlim @ A @ B @ F10 @ F9 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_3688_filterlim__principal,axiom,
! [B: $tType,A: $tType,F2: A > B,S: set @ B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( principal @ B @ S ) @ F5 )
= ( eventually @ A
@ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ S )
@ F5 ) ) ).
% filterlim_principal
thf(fact_3689_le__principal,axiom,
! [A: $tType,F5: filter @ A,A3: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ ( principal @ A @ A3 ) )
= ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 )
@ F5 ) ) ).
% le_principal
thf(fact_3690_eventually__INF1,axiom,
! [B: $tType,A: $tType,I2: A,I: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( member @ A @ I2 @ I )
=> ( ( eventually @ B @ P @ ( F5 @ I2 ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I ) ) ) ) ) ).
% eventually_INF1
thf(fact_3691_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,S2: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F5 @ ( principal @ A @ S2 ) ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ( P @ X2 ) )
@ F5 ) ) ).
% eventually_inf_principal
thf(fact_3692_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( ? [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
& ( ord_less_eq @ ( set @ A ) @ X7 @ A3 )
& ! [Y10: set @ A] :
( ( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X7 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A3 ) )
=> ( P @ Y10 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_3693_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B3 @ A4 )
| ( ord_less_eq @ A @ D3 @ C2 )
| ( ord_less_eq @ A @ B3 @ C2 )
| ( ord_less_eq @ A @ D3 @ A4 ) ) ) ) ).
% Ioc_disjoint
thf(fact_3694_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_3695_eventually__Inf__base,axiom,
! [A: $tType,B5: set @ ( filter @ A ),P: A > $o] :
( ( B5
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B5 )
=> ! [G7: filter @ A] :
( ( member @ ( filter @ A ) @ G7 @ B5 )
=> ? [X6: filter @ A] :
( ( member @ ( filter @ A ) @ X6 @ B5 )
& ( ord_less_eq @ ( filter @ A ) @ X6 @ ( inf_inf @ ( filter @ A ) @ F6 @ G7 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B5 )
& ( eventually @ A @ P @ X2 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_3696_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_3697_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_3698_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_3699_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_3700_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_3701_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ A3 ) ) )
= ( ? [Q: A > B > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( eventually @ B @ ( Q @ X2 ) @ ( F5 @ X2 ) ) )
& ! [Y3: B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( Q @ X2 @ Y3 ) )
=> ( P @ Y3 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_3702_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F2: A > ( set @ B ),A3: set @ B,F5: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F2 @ ( finite5375528669736107172at_top @ B @ A3 ) @ F5 )
= ( ! [X7: set @ B] :
( ( ( finite_finite2 @ B @ X7 )
& ( ord_less_eq @ ( set @ B ) @ X7 @ A3 ) )
=> ( eventually @ A
@ ^ [Y3: A] :
( ( finite_finite2 @ B @ ( F2 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ X7 @ ( F2 @ Y3 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ Y3 ) @ A3 ) )
@ F5 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_3703_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_3704_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_3705_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_3706_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_3707_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ Z9 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_3708_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.head
thf(fact_3709_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: B > ( filter @ A ),B5: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ B5 ) ) )
= ( ? [X7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X7 @ B5 )
& ( finite_finite2 @ B @ X7 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ X7 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_3710_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_3711_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A8: A,B6: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A8 @ B6 ) @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_3712_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_3713_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B5: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X7: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X7 @ B5 )
& ( finite_finite2 @ ( filter @ A ) @ X7 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X7 ) ) ) ) ) ).
% eventually_Inf
thf(fact_3714_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_3715_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A13: set @ B,B16: set @ A,F22: C > D,B24: set @ C,A23: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A13 )
= B16 )
=> ( ( inj_on @ C @ D @ F22 @ B24 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B24 ) @ A23 )
=> ( ( ( B24
= ( bot_bot @ ( set @ C ) ) )
=> ( A23
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B24 @ B16 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B24 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A23 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3716_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A6: A,B2: A,X3: A] :
( ( member @ A @ A6 @ S )
=> ( ( member @ A @ B2 @ S )
=> ( ( ord_less_eq @ A @ A6 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ( member @ A @ X3 @ S ) ) ) ) )
=> ? [A6: A,B2: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B2 ) )
| ( S
= ( set_ord_atMost @ A @ B2 ) )
| ( S
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S
= ( set_ord_atLeast @ A @ A6 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A6 @ B2 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A6 @ B2 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A6 @ B2 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A6 @ B2 ) ) ) ) ) ).
% interval_cases
thf(fact_3717_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y5: set @ B,X4: set @ A,F5: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ Y5 ) @ X4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ Y5 )
@ F5 )
=> ( ( map_filter_on @ A @ C @ X4 @ F2 @ ( map_filter_on @ B @ A @ Y5 @ G @ F5 ) )
= ( map_filter_on @ B @ C @ Y5 @ ( comp @ A @ C @ B @ F2 @ G ) @ F5 ) ) ) ) ).
% map_filter_on_comp
thf(fact_3718_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 ) )
=> ! [Q7: A,R: A] :
( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( R
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= Q7 )
=> ( ( ( modulo_modulo @ A @ A4 @ B3 )
= R )
=> ( A4
!= ( plus_plus @ A @ ( times_times @ A @ Q7 @ B3 ) @ R ) ) ) ) ) ) ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_3719_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_3720_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_3721_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_3722_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_3723_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_3724_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_3725_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_3726_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_3727_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_3728_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_3729_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X4: set @ A,F5: filter @ A,P: B > $o,F2: A > B] :
( ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X4 )
@ F5 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X4 @ F2 @ F5 ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( P @ ( F2 @ X2 ) )
& ( member @ A @ X2 @ X4 ) )
@ F5 ) ) ) ).
% eventually_map_filter_on
thf(fact_3730_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_3731_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N2 ) @ ( set_ord_atLeast @ A @ N2 ) )
= ( insert2 @ A @ N2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_3732_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_3733_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_3734_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R2 ) @ B3 )
= Q4 ) ) ) ) ) ).
% div_bounded
thf(fact_3735_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q4: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R2 )
= A4 )
=> ( ( divide_divide @ A @ A4 @ B3 )
= Q4 ) ) ) ) ) ) ).
% div_eqI
thf(fact_3736_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q4: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R2 )
= A4 )
=> ( ( modulo_modulo @ A @ A4 @ B3 )
= R2 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_3737_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K5: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K5 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_3738_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K5: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K5 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_3739_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_3740_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_3741_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_3742_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_3743_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_3744_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_3745_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N7 @ N4 )
=> ( P @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_3746_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( suc @ X2 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_3747_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N7: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N7 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_3748_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_3749_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_3750_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N7 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_3751_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_3752_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N4: A] :
( ( ord_less @ A @ N7 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_3753_filterlim__atMost__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_atMost_at_top
thf(fact_3754_filterlim__lessThan__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_lessThan_at_top
thf(fact_3755_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_3756_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_3757_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q2: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X3: A,Y2: A] :
( ( Q2 @ X3 )
=> ( ( Q2 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q2 @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q2 @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_3758_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F5 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_3759_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_3760_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_3761_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z9 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_3762_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J: set @ A,I: set @ B,F2: D > C,F5: B > ( filter @ D ),G5: A > ( filter @ C )] :
( ! [M3: A] :
( ( member @ A @ M3 @ J )
=> ? [X6: B] :
( ( member @ B @ X6 @ I )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F2 @ ( F5 @ X6 ) ) @ ( G5 @ M3 ) ) ) )
=> ( filterlim @ D @ C @ F2 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G5 @ J ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F5 @ I ) ) ) ) ).
% filterlim_INF_INF
thf(fact_3763_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [A8: A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert2 @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_3764_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_3765_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F2
@ ^ [X2: A] : ( member @ A @ X2 @ S )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_3766_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_3767_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X2: A] : X2 )
= ( ^ [F7: filter @ A] : F7 ) ) ).
% filtermap_id'
thf(fact_3768_filtermap__id,axiom,
! [A: $tType] :
( ( filtermap @ A @ A @ ( id @ A ) )
= ( id @ ( filter @ A ) ) ) ).
% filtermap_id
thf(fact_3769_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_3770_filtermap__principal,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( filtermap @ B @ A @ F2 @ ( principal @ B @ A3 ) )
= ( principal @ A @ ( image2 @ B @ A @ F2 @ A3 ) ) ) ).
% filtermap_principal
thf(fact_3771_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N4: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_3772_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,F5: filter @ C] :
( ( filtermap @ B @ A @ F2 @ ( filtermap @ C @ B @ G @ F5 ) )
= ( filtermap @ C @ A
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ F5 ) ) ).
% filtermap_filtermap
thf(fact_3773_filtermap__ident,axiom,
! [A: $tType,F5: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X2: A] : X2
@ F5 )
= F5 ) ).
% filtermap_ident
thf(fact_3774_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_3775_filtermap__mono,axiom,
! [B: $tType,A: $tType,F5: filter @ A,F9: filter @ A,F2: A > B] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F9 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F2 @ F5 ) @ ( filtermap @ A @ B @ F2 @ F9 ) ) ) ).
% filtermap_mono
thf(fact_3776_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_3777_filtermap__sup,axiom,
! [A: $tType,B: $tType,F2: B > A,F13: filter @ B,F23: filter @ B] :
( ( filtermap @ B @ A @ F2 @ ( sup_sup @ ( filter @ B ) @ F13 @ F23 ) )
= ( sup_sup @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ F13 ) @ ( filtermap @ B @ A @ F2 @ F23 ) ) ) ).
% filtermap_sup
thf(fact_3778_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,F5: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F2 @ F5 ) )
= ( eventually @ B
@ ^ [X2: B] : ( P @ ( F2 @ X2 ) )
@ F5 ) ) ).
% eventually_filtermap
thf(fact_3779_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,F13: filter @ B,G: C > A,F23: filter @ C] :
( ( filterlim @ A @ B @ F2 @ F13 @ ( filtermap @ C @ A @ G @ F23 ) )
= ( filterlim @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ F13
@ F23 ) ) ).
% filterlim_filtermap
thf(fact_3780_filtermap__eq__strong,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ A,G5: filter @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( filtermap @ A @ B @ F2 @ F5 )
= ( filtermap @ A @ B @ F2 @ G5 ) )
= ( F5 = G5 ) ) ) ).
% filtermap_eq_strong
thf(fact_3781_filterlim__def,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F25: filter @ B,F15: filter @ A] : ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F4 @ F15 ) @ F25 ) ) ) ).
% filterlim_def
thf(fact_3782_map__filter__on__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ B ) ) ).
% map_filter_on_UNIV
thf(fact_3783_filtermap__inf,axiom,
! [A: $tType,B: $tType,F2: B > A,F13: filter @ B,F23: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ ( inf_inf @ ( filter @ B ) @ F13 @ F23 ) ) @ ( inf_inf @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ F13 ) @ ( filtermap @ B @ A @ F2 @ F23 ) ) ) ).
% filtermap_inf
thf(fact_3784_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G: A > B,F5: filter @ B,G5: filter @ A,F2: B > A] :
( ( filterlim @ A @ B @ G @ F5 @ G5 )
=> ( ( filterlim @ B @ A @ F2 @ G5 @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ ( G @ X2 ) )
= X2 )
@ G5 )
=> ( ( filtermap @ B @ A @ F2 @ F5 )
= G5 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_3785_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,F5: C > ( filter @ B ),B5: set @ C] :
( ( filtermap @ B @ A @ F2 @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B5 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B6: C] : ( filtermap @ B @ A @ F2 @ ( F5 @ B6 ) )
@ B5 ) ) ) ).
% filtermap_SUP
thf(fact_3786_filtermap__mono__strong,axiom,
! [B: $tType,A: $tType,F2: A > B,F5: filter @ A,G5: filter @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F2 @ F5 ) @ ( filtermap @ A @ B @ F2 @ G5 ) )
= ( ord_less_eq @ ( filter @ A ) @ F5 @ G5 ) ) ) ).
% filtermap_mono_strong
thf(fact_3787_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( filtermap @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F2 ) @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( finite5375528669736107172at_top @ B @ ( image2 @ A @ B @ F2 @ A3 ) ) ) ) ).
% filtermap_image_finite_subsets_at_top
thf(fact_3788_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,F5: C > ( filter @ B ),B5: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B5 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B6: C] : ( filtermap @ B @ A @ F2 @ ( F5 @ B6 ) )
@ B5 ) ) ) ).
% filtermap_INF
thf(fact_3789_pairwise__disjnt__iff,axiom,
! [A: $tType,A14: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A14 )
= ( ! [X2: A] :
( uniq @ ( set @ A )
@ ^ [X7: set @ A] :
( ( member @ ( set @ A ) @ X7 @ A14 )
& ( member @ A @ X2 @ X7 ) ) ) ) ) ).
% pairwise_disjnt_iff
thf(fact_3790_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_3791_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_3792_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 )
@ ^ [A8: A] : ( product_Pair @ A @ B @ A8 @ X )
@ F5 ) ) ).
% prod_filter_principal_singleton2
thf(fact_3793_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,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% plus_rat_def
thf(fact_3794_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F5: filter @ A,G5: filter @ B,H9: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F5 @ G5 ) @ H9 )
= ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [X2: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y3: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G5 @ H9 ) ) ) ) ).
% prod_filter_assoc
thf(fact_3795_prod__filter__mono,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F9: filter @ A,G5: filter @ B,G6: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F9 )
=> ( ( ord_less_eq @ ( filter @ B ) @ G5 @ G6 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ F5 @ G5 ) @ ( prod_filter @ A @ B @ F9 @ G6 ) ) ) ) ).
% prod_filter_mono
thf(fact_3796_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B5: filter @ B] :
( ( ( prod_filter @ A @ B @ A3 @ B5 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( filter @ A ) ) )
| ( B5
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_3797_prod__filter__commute,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F7: filter @ A,G8: filter @ B] : ( filtermap @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( prod_filter @ B @ A @ G8 @ F7 ) ) ) ) ).
% prod_filter_commute
thf(fact_3798_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F5: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F5 @ F5 ) )
= ( ? [Q: A > $o] :
( ( eventually @ A @ Q @ F5 )
& ! [X2: A,Y3: A] :
( ( Q @ X2 )
=> ( ( Q @ Y3 )
=> ( P @ ( product_Pair @ A @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_3799_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F5: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F5 @ G5 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F5 )
& ( eventually @ B @ Pg @ G5 )
& ! [X2: A,Y3: B] :
( ( Pf @ X2 )
=> ( ( Pg @ Y3 )
=> ( P @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_3800_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A3: filter @ A,B5: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( prod_filter @ A @ B @ A3 @ B5 ) ) @ A3 ) ).
% filtermap_fst_prod_filter
thf(fact_3801_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A3: filter @ B,B5: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( prod_filter @ B @ A @ A3 @ B5 ) ) @ B5 ) ).
% filtermap_snd_prod_filter
thf(fact_3802_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B5: filter @ B,C6: filter @ A,D4: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A3 @ B5 ) @ ( prod_filter @ A @ B @ C6 @ D4 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A3 @ C6 )
& ( ord_less_eq @ ( filter @ B ) @ B5 @ D4 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_3803_prod__filtermap1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,F5: filter @ C,G5: filter @ B] :
( ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F2 @ F5 ) @ G5 )
= ( filtermap @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ ( product_apfst @ C @ A @ B @ F2 ) @ ( prod_filter @ C @ B @ F5 @ G5 ) ) ) ).
% prod_filtermap1
thf(fact_3804_prod__filtermap2,axiom,
! [B: $tType,A: $tType,C: $tType,F5: filter @ A,G: C > B,G5: filter @ C] :
( ( prod_filter @ A @ B @ F5 @ ( filtermap @ C @ B @ G @ G5 ) )
= ( filtermap @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B ) @ ( product_apsnd @ C @ B @ A @ G ) @ ( prod_filter @ A @ C @ F5 @ G5 ) ) ) ).
% prod_filtermap2
thf(fact_3805_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G5: filter @ B,F5: filter @ A,G: A > C,H9: filter @ C] :
( ( filterlim @ A @ B @ F2 @ G5 @ F5 )
=> ( ( filterlim @ A @ C @ G @ H9 @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( prod_filter @ B @ C @ G5 @ H9 )
@ F5 ) ) ) ).
% filterlim_Pair
thf(fact_3806_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > A,G: 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 ) @ ( G @ X2 ) )
@ F5 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F2 @ F5 ) @ ( filtermap @ C @ B @ G @ F5 ) ) ) ).
% filtermap_Pair
thf(fact_3807_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A3 ) @ ( principal @ B @ B5 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) ) ).
% principal_prod_principal
thf(fact_3808_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_3809_bounded__quasi__semilattice__set_Oinsert,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 @ A3 ) ) ) ) ).
% bounded_quasi_semilattice_set.insert
thf(fact_3810_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F7: filter @ A,G8: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P2: A > $o,Q: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y3: B] :
( ( P2 @ X2 )
& ( Q @ Y3 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P2: A > $o,Q: B > $o] :
( ( eventually @ A @ P2 @ F7 )
& ( eventually @ B @ Q @ G8 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_3811_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F5: filter @ ( product_prod @ A @ B ),A3: filter @ A,B5: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ F5 ) @ A3 )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ F5 ) @ B5 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ F5 @ ( prod_filter @ A @ B @ A3 @ B5 ) ) ) ) ).
% le_prod_filterI
thf(fact_3812_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 ) ) )
= ( ? [N7: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ N7 @ M4 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ N7 @ N4 )
=> ( P @ ( product_Pair @ nat @ nat @ N4 @ M4 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_3813_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: filter @ A,Q2: B > $o,G5: filter @ B] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ B @ Q2 @ G5 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X2 ) )
& ( Q2 @ ( product_snd @ A @ B @ X2 ) ) )
@ ( prod_filter @ A @ B @ F5 @ G5 ) ) ) ) ).
% eventually_prodI
thf(fact_3814_eventually__prod1,axiom,
! [A: $tType,B: $tType,B5: filter @ A,P: B > $o,A3: filter @ B] :
( ( B5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X2: B,Y3: A] : ( P @ X2 ) )
@ ( prod_filter @ B @ A @ A3 @ B5 ) )
= ( eventually @ B @ P @ A3 ) ) ) ).
% eventually_prod1
thf(fact_3815_eventually__prod2,axiom,
! [A: $tType,B: $tType,A3: filter @ A,P: B > $o,B5: 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 @ B5 ) )
= ( eventually @ B @ P @ B5 ) ) ) ).
% eventually_prod2
thf(fact_3816_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I: set @ A,J: set @ B,A3: A > ( filter @ C ),B5: B > ( filter @ D )] :
( ( I
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A3 @ I ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B5 @ J ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I4: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A3 @ I4 ) @ ( B5 @ J3 ) )
@ J ) )
@ I ) ) ) ) ) ).
% prod_filter_INF
thf(fact_3817_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I: set @ A,A3: A > ( filter @ B ),B5: filter @ C] :
( ( I
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A3 @ I ) ) @ B5 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ ( A3 @ I4 ) @ B5 )
@ I ) ) ) ) ).
% prod_filter_INF1
thf(fact_3818_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J: set @ A,A3: filter @ B,B5: A > ( filter @ C )] :
( ( J
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B5 @ J ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ A3 @ ( B5 @ I4 ) )
@ J ) ) ) ) ).
% prod_filter_INF2
thf(fact_3819_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,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% times_rat_def
thf(fact_3820_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_3821_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ B ),F4: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A15: B,A24: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F4 @ A15 ) @ ( F4 @ A24 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A15 @ A24 ) @ R3 ) ) ) ) ) ).
% relImage_def
thf(fact_3822_add__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A4 @ B3 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A4 @ D3 ) @ ( times_times @ int @ C2 @ B3 ) ) @ ( times_times @ int @ B3 @ D3 ) ) ) ) ) ).
% add_rat
thf(fact_3823_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: C > A,I: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A3 @ I ) )
=> ( ( I
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A3 @ I ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3824_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_3825_divide__rat,axiom,
! [A4: int,B3: int,C2: int,D3: int] :
( ( divide_divide @ rat @ ( fract @ A4 @ B3 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( times_times @ int @ A4 @ D3 ) @ ( times_times @ int @ B3 @ C2 ) ) ) ).
% divide_rat
thf(fact_3826_sgn__rat,axiom,
! [A4: int,B3: int] :
( ( sgn_sgn @ rat @ ( fract @ A4 @ B3 ) )
= ( ring_1_of_int @ rat @ ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ ( sgn_sgn @ int @ B3 ) ) ) ) ).
% sgn_rat
thf(fact_3827_mult__rat,axiom,
! [A4: int,B3: int,C2: int,D3: int] :
( ( times_times @ rat @ ( fract @ A4 @ B3 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( times_times @ int @ A4 @ C2 ) @ ( times_times @ int @ B3 @ D3 ) ) ) ).
% mult_rat
thf(fact_3828_le__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A4 @ B3 ) @ ( fract @ C2 @ D3 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A4 @ D3 ) @ ( times_times @ int @ B3 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B3 ) @ ( times_times @ int @ B3 @ D3 ) ) ) ) ) ) ).
% le_rat
thf(fact_3829_less__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A4 @ B3 ) @ ( fract @ C2 @ D3 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A4 @ D3 ) @ ( times_times @ int @ B3 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B3 ) @ ( times_times @ int @ B3 @ D3 ) ) ) ) ) ) ).
% less_rat
thf(fact_3830_diff__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( minus_minus @ rat @ ( fract @ A4 @ B3 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( minus_minus @ int @ ( times_times @ int @ A4 @ D3 ) @ ( times_times @ int @ C2 @ B3 ) ) @ ( times_times @ int @ B3 @ D3 ) ) ) ) ) ).
% diff_rat
thf(fact_3831_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A4 ) ) ) ).
% mono_add
thf(fact_3832_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_3833_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_3834_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_3835_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% monoI
thf(fact_3836_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( F4 @ Y3 ) ) ) ) ) ) ).
% mono_def
thf(fact_3837_mono__Int,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A3: set @ A,B5: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ B ) @ ( F2 @ A3 ) @ ( F2 @ B5 ) ) ) ) ).
% mono_Int
thf(fact_3838_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_3839_mono__Un,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A3: set @ A,B5: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F2 @ A3 ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% mono_Un
thf(fact_3840_eq__rat_I1_J,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ( fract @ A4 @ B3 )
= ( fract @ C2 @ D3 ) )
= ( ( times_times @ int @ A4 @ D3 )
= ( times_times @ int @ C2 @ B3 ) ) ) ) ) ).
% eq_rat(1)
thf(fact_3841_mult__rat__cancel,axiom,
! [C2: int,A4: int,B3: int] :
( ( C2
!= ( zero_zero @ int ) )
=> ( ( fract @ ( times_times @ int @ C2 @ A4 ) @ ( times_times @ int @ C2 @ B3 ) )
= ( fract @ A4 @ B3 ) ) ) ).
% mult_rat_cancel
thf(fact_3842_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_3843_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_3844_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ? [X6: A] :
( ( member @ A @ X6 @ S )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S )
=> ( ord_less_eq @ A @ X6 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y3: B] : ( member @ B @ Y3 @ ( image2 @ A @ B @ F2 @ S ) ) )
= ( F2
@ ( ord_Least @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_3845_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ 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_3846_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ 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_3847_positive__rat,axiom,
! [A4: int,B3: int] :
( ( positive @ ( fract @ A4 @ B3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A4 @ B3 ) ) ) ).
% positive_rat
thf(fact_3848_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A5: set @ A,A8: A] :
( ( X2
= ( insert2 @ A @ A8 @ A5 ) )
& ( P5 @ A5 ) ) ) ) ).
% finite.mono
thf(fact_3849_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_3850_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A3: C > A,I: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A3 @ I ) )
=> ( ( I
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A3 @ I ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3851_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,F5: filter @ A,G5: filter @ B] :
( ( rel_filter @ A @ B @ R4 @ F5 @ G5 )
=> ~ ! [Z10: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) @ Z10 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_fst @ A @ B ) @ Z10 )
= F5 )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_snd @ A @ B ) @ Z10 )
!= G5 ) ) ) ) ).
% rel_filter.cases
thf(fact_3852_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R3: A > B > $o,F7: filter @ A,G8: filter @ B] :
? [Z9: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) @ Z9 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_fst @ A @ B ) @ Z9 )
= F7 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_snd @ A @ B ) @ Z9 )
= G8 ) ) ) ) ).
% rel_filter.simps
thf(fact_3853_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,Z6: filter @ ( product_prod @ A @ B ),F5: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) @ Z6 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_fst @ A @ B ) @ Z6 )
= F5 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_snd @ A @ B ) @ Z6 )
= G5 )
=> ( rel_filter @ A @ B @ R4 @ F5 @ G5 ) ) ) ) ).
% rel_filter.intros
thf(fact_3854_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q2: A > B > C,F2: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q2 )
=> ( order_mono @ A @ ( D > C )
@ ^ [I4: A,X2: D] : ( Q2 @ I4 @ ( F2 @ X2 ) ) ) ) ) ).
% mono_compose
thf(fact_3855_rel__filter__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B5 )
=> ( ord_less_eq @ ( ( filter @ A ) > ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ B5 ) ) ) ).
% rel_filter_mono
thf(fact_3856_rel__filter__eq,axiom,
! [A: $tType] :
( ( rel_filter @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [Y4: filter @ A,Z5: filter @ A] : Y4 = Z5 ) ) ).
% rel_filter_eq
thf(fact_3857_bot__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( rel_filter @ A @ B @ A3 @ ( bot_bot @ ( filter @ A ) ) @ ( bot_bot @ ( filter @ B ) ) ) ).
% bot_filter_parametric
thf(fact_3858_sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( sup_sup @ ( filter @ A ) ) @ ( sup_sup @ ( filter @ B ) ) ) ).
% sup_filter_parametric
thf(fact_3859_eventually__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ( eventually @ A )
@ ( eventually @ B ) ) ).
% eventually_parametric
thf(fact_3860_filtermap__parametric,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: A > C > $o,B5: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( filter @ A ) > ( filter @ B ) ) @ ( ( filter @ C ) > ( filter @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A3 @ B5 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( filter @ D ) @ ( rel_filter @ A @ C @ A3 ) @ ( rel_filter @ B @ D @ B5 ) ) @ ( filtermap @ A @ B ) @ ( filtermap @ C @ D ) ) ).
% filtermap_parametric
thf(fact_3861_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: set @ A,B5: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A3 @ B5 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) ) ).
% member_product
thf(fact_3862_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A5: set @ A,B8: set @ B] :
( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B8 ) ) ) ).
% Product_Type.product_def
thf(fact_3863_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] :
! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_3864_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] :
! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_3865_ball__empty,axiom,
! [A: $tType,P: A > $o,X6: A] :
( ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X6 ) ) ).
% ball_empty
thf(fact_3866_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_3867_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P2: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( P2 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_3868_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,G: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F2 @ A3 ) )
=> ( G @ X2 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( comp @ A @ $o @ B @ G @ F2 @ X2 ) ) ) ) ).
% Ball_image_comp
thf(fact_3869_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B
@ ^ [X2: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( P @ X2 @ Y3 ) )
@ Net )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X2 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_3870_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( eventually @ B
@ ^ [Y3: B] : ( P @ Y3 @ X3 )
@ Net ) )
=> ( eventually @ B
@ ^ [X2: B] :
! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( P @ X2 @ Y3 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_3871_Field__not__elem,axiom,
! [A: $tType,V: A,R4: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ V @ ( field2 @ A @ R4 ) )
=> ! [X6: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X6 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z3: A] :
( ( Y3 != V )
& ( Z3 != V ) )
@ X6 ) ) ) ).
% Field_not_elem
thf(fact_3872_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A8: A,B6: A] : A8 != B6
@ X2 ) ) ) ) ).
% irrefl_distinct
thf(fact_3873_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ C4 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_3874_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Y13: A] :
! [Z3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z3 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y13 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa3 ) @ R5 ) )
@ Z3 ) )
@ X2 ) ) ) ) ).
% trans_join
thf(fact_3875_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A5: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z3: A] :
( ( member @ A @ Y3 @ A5 )
& ( member @ A @ Z3 @ A5 ) )
@ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_3876_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S6: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F7: filter @ A] :
! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S6 )
=> ( ord_less_eq @ ( filter @ A ) @ F7 @ X2 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_3877_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( B6 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ X2 ) @ R5 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_3878_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ X2 ) @ R5 ) ) ) ) ) ) ).
% Under_def
thf(fact_3879_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B6 ) @ R5 ) ) ) ) ) ) ).
% Above_def
thf(fact_3880_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R5: 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 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_3881_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X6: A] :
( ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_3882_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% bex_UNIV
thf(fact_3883_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o,A3: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) @ A3 )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 @ Y3 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_3884_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F4: A > B,A5: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( Y3
= ( F4 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_3885_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A5: set @ A,P2: A > $o] :
? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_3886_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S7: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X2: A] :
( ( member @ A @ X2 @ S7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% Image_def
thf(fact_3887_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( vimage @ A @ B @ F2 @ ( image2 @ A @ B @ F2 @ A3 ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( F2 @ Y3 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_3888_max__extp_Omax__extI,axiom,
! [A: $tType,X4: set @ A,Y5: set @ A,R4: A > A > $o] :
( ( finite_finite2 @ A @ X4 )
=> ( ( finite_finite2 @ A @ Y5 )
=> ( ( Y5
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y5 )
& ( R4 @ X3 @ Xa2 ) ) )
=> ( max_extp @ A @ R4 @ X4 @ Y5 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_3889_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R3: A > A > $o,A15: set @ A,A24: set @ A] :
( ( finite_finite2 @ A @ A15 )
& ( finite_finite2 @ A @ A24 )
& ( A24
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A15 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A24 )
& ( R3 @ X2 @ Y3 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_3890_max__extp_Ocases,axiom,
! [A: $tType,R4: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R4 @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X6: A] :
( ( member @ A @ X6 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R4 @ X6 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_3891_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R3: 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_finite2 @ A @ X7 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_3892_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S6: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P2: A > $o] :
! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S6 )
=> ( eventually @ A @ P2 @ X2 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_3893_frequently__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ( frequently @ A )
@ ( frequently @ B ) ) ).
% frequently_parametric
thf(fact_3894_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( B6 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B6 ) @ R5 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_3895_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_3896_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X7: A] : ( P @ X7 ) ) ) ).
% ball_UNIV
thf(fact_3897_frequently__mono,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ( frequently @ A @ P @ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_mono
thf(fact_3898_frequentlyE,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( frequently @ A @ P @ F5 )
=> ~ ! [X3: A] :
~ ( P @ X3 ) ) ).
% frequentlyE
thf(fact_3899_not__frequently__False,axiom,
! [A: $tType,F5: filter @ A] :
~ ( frequently @ A
@ ^ [X2: A] : $false
@ F5 ) ).
% not_frequently_False
thf(fact_3900_frequently__disj__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q2 @ X2 ) )
@ F5 )
= ( ( frequently @ A @ P @ F5 )
| ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_disj_iff
thf(fact_3901_frequently__elim1,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( frequently @ A @ P @ F5 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( Q2 @ I3 ) )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_elim1
thf(fact_3902_frequently__disj,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( frequently @ A @ P @ F5 )
=> ( ( frequently @ A @ Q2 @ F5 )
=> ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q2 @ X2 ) )
@ F5 ) ) ) ).
% frequently_disj
thf(fact_3903_frequently__all,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
! [X7: B] : ( P @ X2 @ X7 )
@ F5 )
= ( ! [Y10: A > B] :
( frequently @ A
@ ^ [X2: A] : ( P @ X2 @ ( Y10 @ X2 ) )
@ F5 ) ) ) ).
% frequently_all
thf(fact_3904_frequently__ex,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( frequently @ A @ P @ F5 )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% frequently_ex
thf(fact_3905_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
& C6 )
@ F5 )
= ( ( frequently @ A @ P @ F5 )
& C6 ) ) ).
% eventually_frequently_const_simps(1)
thf(fact_3906_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( C6
& ( P @ X2 ) )
@ F5 )
= ( C6
& ( frequently @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(2)
thf(fact_3907_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P6: A > $o] :
! [X9: A] : ( P6 @ X9 ) ) ) ).
% top_filter_def
thf(fact_3908_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y2: B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ Y2 )
@ Net )
=> ( eventually @ A
@ ^ [X2: A] :
! [X7: B] : ( P @ X2 @ X7 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_3909_frequently__imp__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_imp_iff
thf(fact_3910_frequently__rev__mp,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( frequently @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
@ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_rev_mp
thf(fact_3911_not__frequently,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( ~ ( frequently @ A @ P @ F5 ) )
= ( eventually @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ F5 ) ) ).
% not_frequently
thf(fact_3912_not__eventually,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ( ~ ( eventually @ A @ P @ F5 ) )
= ( frequently @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ F5 ) ) ).
% not_eventually
thf(fact_3913_frequently__def,axiom,
! [A: $tType] :
( ( frequently @ A )
= ( ^ [P2: A > $o,F7: filter @ A] :
~ ( eventually @ A
@ ^ [X2: A] :
~ ( P2 @ X2 )
@ F7 ) ) ) ).
% frequently_def
thf(fact_3914_frequently__mp,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
@ F5 )
=> ( ( frequently @ A @ P @ F5 )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ).
% frequently_mp
thf(fact_3915_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> C6 )
@ F5 )
= ( ( frequently @ A @ P @ F5 )
=> C6 ) ) ).
% eventually_frequently_const_simps(5)
thf(fact_3916_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_3917_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
( abs_filter @ A
@ ^ [P2: A > $o] :
? [Q: A > $o,R3: A > $o] :
( ( eventually @ A @ Q @ F7 )
& ( eventually @ A @ R3 @ F8 )
& ! [X2: A] :
( ( ( Q @ X2 )
& ( R3 @ X2 ) )
=> ( P2 @ X2 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_3918_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A5: A > C > $o,B8: B > D > $o,F4: A > B,G4: C > D] :
! [X2: A,Y3: C] :
( ( A5 @ X2 @ Y3 )
=> ( B8 @ ( F4 @ X2 ) @ ( G4 @ Y3 ) ) ) ) ) ).
% rel_fun_def
thf(fact_3919_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P2: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_3920_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_3921_frequently__bex__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,F5: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( frequently @ B
@ ^ [X2: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A3 )
& ( P @ X2 @ Y3 ) )
@ F5 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X2 )
@ F5 ) ) ) ) ) ).
% frequently_bex_finite_distrib
thf(fact_3922_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,F5: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( frequently @ B
@ ^ [X2: B] :
? [Y3: A] :
( ( member @ A @ Y3 @ A3 )
& ( P @ X2 @ Y3 ) )
@ F5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( frequently @ B
@ ^ [Y3: B] : ( P @ Y3 @ X3 )
@ F5 ) ) ) ) ).
% frequently_bex_finite
thf(fact_3923_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ~ ( P @ X2 )
| ( Q2 @ X2 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q2 @ F5 ) ) ) ) ).
% eventually_frequentlyE
thf(fact_3924_AboveS__disjoint,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( order_AboveS @ A @ R2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_3925_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
! [Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( P @ Y3 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_3926_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P2 @ X2 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_3927_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
( abs_filter @ A
@ ^ [P2: A > $o] :
( ( eventually @ A @ P2 @ F7 )
& ( eventually @ A @ P2 @ F8 ) ) ) ) ) ).
% sup_filter_def
thf(fact_3928_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S6: set @ A] : ( abs_filter @ A @ ( ball @ A @ S6 ) ) ) ) ).
% principal_def
thf(fact_3929_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F4: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P2: B > $o] :
( eventually @ A
@ ^ [X2: A] : ( P2 @ ( F4 @ X2 ) )
@ F7 ) ) ) ) ).
% filtermap_def
thf(fact_3930_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X7: set @ A,F4: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P2: B > $o] :
( eventually @ A
@ ^ [X2: A] :
( ( P2 @ ( F4 @ X2 ) )
& ( member @ A @ X2 @ X7 ) )
@ F7 ) ) ) ) ).
% map_filter_on_def
thf(fact_3931_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P2 @ X2 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_3932_wo__rel_Osuc__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B3 @ B5 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 )
!= B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_3933_wo__rel_Osuc__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_3934_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) @ ( order_AboveS @ A @ R2 @ B5 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_3935_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_3936_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_3937_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( order_AboveS @ A @ R2 @ B5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) @ A4 ) @ R2 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_3938_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ A,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A4 @ ( order_AboveS @ A @ R2 @ B5 ) )
=> ( ! [A19: A] :
( ( member @ A @ A19 @ ( order_AboveS @ A @ R2 @ B5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A19 ) @ R2 ) )
=> ( A4
= ( bNF_Wellorder_wo_suc @ A @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_3939_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P2: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P2 @ X2 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Less_eq2 @ X2 @ Y3 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_3940_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( ( order_AboveS @ A @ R2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ A3 ) ) @ R2 )
=> ( ( B3
!= ( bNF_Wellorder_wo_suc @ A @ R2 @ A3 ) )
=> ( member @ A @ B3 @ A3 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_3941_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F4: A > B,F7: filter @ B] :
( abs_filter @ A
@ ^ [P2: A > $o] :
? [Q: B > $o] :
( ( eventually @ B @ Q @ F7 )
& ! [X2: A] :
( ( Q @ ( F4 @ X2 ) )
=> ( P2 @ X2 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_3942_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
@ ^ [R5: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S7 )
& ! [A8: A,B6: A,C5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ S7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R5 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_3943_filterlim__filtercomap,axiom,
! [A: $tType,B: $tType,F2: A > B,F5: filter @ B] : ( filterlim @ A @ B @ F2 @ F5 @ ( filtercomap @ A @ B @ F2 @ F5 ) ) ).
% filterlim_filtercomap
thf(fact_3944_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_3945_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F5: filter @ A,F2: B > A] :
( ( eventually @ A @ P @ F5 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F2 @ X2 ) )
@ ( filtercomap @ B @ A @ F2 @ F5 ) ) ) ).
% eventually_filtercomapI
thf(fact_3946_filtercomap__top,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( filtercomap @ A @ B @ F2 @ ( top_top @ ( filter @ B ) ) )
= ( top_top @ ( filter @ A ) ) ) ).
% filtercomap_top
thf(fact_3947_filtercomap__principal,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ B] :
( ( filtercomap @ A @ B @ F2 @ ( principal @ B @ A3 ) )
= ( principal @ A @ ( vimage @ A @ B @ F2 @ A3 ) ) ) ).
% filtercomap_principal
thf(fact_3948_eventually__filtercomap,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,F5: filter @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F2 @ F5 ) )
= ( ? [Q: B > $o] :
( ( eventually @ B @ Q @ F5 )
& ! [X2: A] :
( ( Q @ ( F2 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap
thf(fact_3949_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_3950_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G: B > C,F5: filter @ C] :
( ( filtercomap @ A @ B @ F2 @ ( filtercomap @ B @ C @ G @ F5 ) )
= ( filtercomap @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ F5 ) ) ).
% filtercomap_filtercomap
thf(fact_3951_filtercomap__ident,axiom,
! [A: $tType,F5: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X2: A] : X2
@ F5 )
= F5 ) ).
% filtercomap_ident
thf(fact_3952_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F5: filter @ A,F9: filter @ A,F2: B > A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F9 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtercomap @ B @ A @ F2 @ F5 ) @ ( filtercomap @ B @ A @ F2 @ F9 ) ) ) ).
% filtercomap_mono
thf(fact_3953_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F2: A > B,F13: filter @ B,F23: filter @ B] :
( ( filtercomap @ A @ B @ F2 @ ( inf_inf @ ( filter @ B ) @ F13 @ F23 ) )
= ( inf_inf @ ( filter @ A ) @ ( filtercomap @ A @ B @ F2 @ F13 ) @ ( filtercomap @ A @ B @ F2 @ F23 ) ) ) ).
% filtercomap_inf
thf(fact_3954_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G: B > C,G5: filter @ C,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( filtercomap @ B @ C @ G @ G5 ) @ F5 )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ G5 @ F5 ) ) ).
% filterlim_filtercomap_iff
thf(fact_3955_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ! [P4: A > $o] :
( ( eventually @ A @ P4 @ F5 )
=> ? [X6: B] : ( P4 @ ( F2 @ X6 ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_3956_filterlim__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F7: filter @ B,G8: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ G8 @ ( filtercomap @ A @ B @ F4 @ F7 ) ) ) ) ).
% filterlim_iff_le_filtercomap
thf(fact_3957_filtercomap__filtermap,axiom,
! [B: $tType,A: $tType,F5: filter @ A,F2: A > B] : ( ord_less_eq @ ( filter @ A ) @ F5 @ ( filtercomap @ A @ B @ F2 @ ( filtermap @ A @ B @ F2 @ F5 ) ) ) ).
% filtercomap_filtermap
thf(fact_3958_filtermap__filtercomap,axiom,
! [B: $tType,A: $tType,F2: B > A,F5: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ ( filtercomap @ B @ A @ F2 @ F5 ) ) @ F5 ) ).
% filtermap_filtercomap
thf(fact_3959_filtermap__le__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType,F2: B > A,F5: filter @ B,G5: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F2 @ F5 ) @ G5 )
= ( ord_less_eq @ ( filter @ B ) @ F5 @ ( filtercomap @ B @ A @ F2 @ G5 ) ) ) ).
% filtermap_le_iff_le_filtercomap
thf(fact_3960_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F2: A > B,F13: filter @ B,F23: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F2 @ F13 ) @ ( filtercomap @ A @ B @ F2 @ F23 ) ) @ ( filtercomap @ A @ B @ F2 @ ( sup_sup @ ( filter @ B ) @ F13 @ F23 ) ) ) ).
% filtercomap_sup
thf(fact_3961_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,F5: C > ( filter @ B ),B5: set @ C] :
( ( filtercomap @ A @ B @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B5 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B6: C] : ( filtercomap @ A @ B @ F2 @ ( F5 @ B6 ) )
@ B5 ) ) ) ).
% filtercomap_INF
thf(fact_3962_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_3963_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N7: A] :
! [X2: B] :
( ( ord_less_eq @ A @ N7 @ ( F2 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_3964_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N7: A] :
! [X2: B] :
( ( ord_less @ A @ N7 @ ( F2 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_3965_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N7: A] :
! [X2: B] :
( ( ord_less_eq @ A @ ( F2 @ X2 ) @ N7 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_3966_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N7: A] :
! [X2: B] :
( ( ord_less @ A @ ( F2 @ X2 ) @ N7 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_3967_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > C,F5: B > ( filter @ C ),B5: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B6: B] : ( filtercomap @ A @ C @ F2 @ ( F5 @ B6 ) )
@ B5 ) )
@ ( filtercomap @ A @ C @ F2 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F5 @ B5 ) ) ) ) ).
% filtercomap_SUP
thf(fact_3968_flip__pred,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ A @ B ),R4: 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 @ R4 ) ) ) )
=> ( 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,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X2 ) )
@ A3 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R4 ) ) ) ) ).
% flip_pred
thf(fact_3969_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_3970_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3971_mset__set_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ 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_3972_conversep__eq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [Y4: A,Z5: A] : Y4 = Z5 ) ) ).
% conversep_eq
thf(fact_3973_conversep__iff,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R5: A > B > $o,A8: B,B6: A] : ( R5 @ B6 @ A8 ) ) ) ).
% conversep_iff
thf(fact_3974_conversep__inject,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S2: B > A > $o] :
( ( ( conversep @ B @ A @ R2 )
= ( conversep @ B @ A @ S2 ) )
= ( R2 = S2 ) ) ).
% conversep_inject
thf(fact_3975_conversep__conversep,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( conversep @ B @ A @ ( conversep @ A @ B @ R2 ) )
= R2 ) ).
% conversep_conversep
thf(fact_3976_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X2: A,Y3: A] : X2 != Y3 )
= ( ^ [X2: A,Y3: A] : X2 != Y3 ) ) ).
% conversep_noteq
thf(fact_3977_conversep__mono,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S2: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S2 ) )
= ( ord_less_eq @ ( B > A > $o ) @ R2 @ S2 ) ) ).
% conversep_mono
thf(fact_3978_mset__set_Oinsert,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( mset_set @ A @ ( insert2 @ A @ X @ A3 ) )
= ( add_mset @ A @ X @ ( mset_set @ A @ A3 ) ) ) ) ) ).
% mset_set.insert
thf(fact_3979_conversep__le__swap,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,S2: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ ( conversep @ B @ A @ S2 ) )
= ( ord_less_eq @ ( B > A > $o ) @ ( conversep @ A @ B @ R2 ) @ S2 ) ) ).
% conversep_le_swap
thf(fact_3980_leq__conversepI,axiom,
! [A: $tType,R4: A > A > $o] :
( ( R4
= ( ^ [Y4: A,Z5: A] : Y4 = Z5 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R4 @ ( conversep @ A @ A @ R4 ) ) ) ).
% leq_conversepI
thf(fact_3981_mset__union__2__elem,axiom,
! [A: $tType,A4: A,B3: A,C2: A,M2: multiset @ A] :
( ( ( add_mset @ A @ A4 @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C2 @ M2 ) )
=> ( ( ( ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) )
= M2 )
& ( B3 = C2 ) )
| ( ( A4 = C2 )
& ( ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) )
= M2 ) ) ) ) ).
% mset_union_2_elem
thf(fact_3982_conversep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R5: A > B > $o,A15: B,A24: A] :
? [A8: A,B6: B] :
( ( A15 = B6 )
& ( A24 = A8 )
& ( R5 @ A8 @ B6 ) ) ) ) ).
% conversep.simps
thf(fact_3983_conversepD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B3: B,A4: A] :
( ( conversep @ A @ B @ R2 @ B3 @ A4 )
=> ( R2 @ A4 @ B3 ) ) ).
% conversepD
thf(fact_3984_conversepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: B,A22: A] :
( ( conversep @ A @ B @ R2 @ A1 @ A22 )
=> ( R2 @ A22 @ A1 ) ) ).
% conversepE
thf(fact_3985_conversepI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,A4: A,B3: B] :
( ( R2 @ A4 @ B3 )
=> ( conversep @ A @ B @ R2 @ B3 @ A4 ) ) ).
% conversepI
thf(fact_3986_single__valuedpD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: A,Y: B,Z2: B] :
( ( single_valuedp @ A @ B @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ X @ Z2 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedpD
thf(fact_3987_single__valuedpI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ! [X3: A,Y2: B,Z4: B] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ X3 @ Z4 )
=> ( Y2 = Z4 ) ) )
=> ( single_valuedp @ A @ B @ R2 ) ) ).
% single_valuedpI
thf(fact_3988_single__valuedp__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A,Y3: B] :
( ( R5 @ X2 @ Y3 )
=> ! [Z3: B] :
( ( R5 @ X2 @ Z3 )
=> ( Y3 = Z3 ) ) ) ) ) ).
% single_valuedp_def
thf(fact_3989_converse__meet,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S2: B > A > $o] :
( ( conversep @ B @ A @ ( inf_inf @ ( B > A > $o ) @ R2 @ S2 ) )
= ( inf_inf @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S2 ) ) ) ).
% converse_meet
thf(fact_3990_converse__join,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S2: B > A > $o] :
( ( conversep @ B @ A @ ( sup_sup @ ( B > A > $o ) @ R2 @ S2 ) )
= ( sup_sup @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S2 ) ) ) ).
% converse_join
thf(fact_3991_rel__filter__conversep,axiom,
! [A: $tType,B: $tType,A3: B > A > $o] :
( ( rel_filter @ A @ B @ ( conversep @ B @ A @ A3 ) )
= ( conversep @ ( filter @ B ) @ ( filter @ A ) @ ( rel_filter @ B @ A @ A3 ) ) ) ).
% rel_filter_conversep
thf(fact_3992_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: B,Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) @ ( converse @ A @ B @ R2 ) ) ) ) ).
% conversep_converse_eq
thf(fact_3993_single__valuedp__less__eq,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,S2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ S2 )
=> ( ( single_valuedp @ A @ B @ S2 )
=> ( single_valuedp @ A @ B @ R2 ) ) ) ).
% single_valuedp_less_eq
thf(fact_3994_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : ( single_valuedp @ A @ B @ ( bot_bot @ ( A > B > $o ) ) ) ).
% single_valuedp_bot
thf(fact_3995_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ) ).
% converse_def
thf(fact_3996_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B5: multiset @ A,C6: multiset @ A,S2: A,A3: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B5 @ C6 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 ) )
=> ( ( ( B5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C6 ) ) )
=> ~ ( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_3997_mset__unplusm__dist__cases,axiom,
! [A: $tType,S2: A,A3: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
= ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) )
=> ( ( ( B5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C6 ) ) )
=> ~ ( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_3998_mset__single__cases2_H,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C10: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C10 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C10 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C10
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C10 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_3999_mset__single__cases2,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C10: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C10 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C10 ) )
=> ~ ( ( C10
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C10 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C10 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_4000_mset__single__cases_H,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C10: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C10 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C10 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C10
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C10 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_4001_mset__single__cases,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C10: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C10 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C10 ) )
=> ~ ( ( C10
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C10 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C10 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_4002_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A] : ( uniq @ B @ ( R5 @ X2 ) ) ) ) ).
% single_valuedp_iff_Uniq
thf(fact_4003_mset__set_Oremove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ 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_4004_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A4: A,M2: multiset @ A] :
( ~ ( member @ A @ A4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M2 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M2 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M2 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_4005_normalize__crossproduct,axiom,
! [Q4: int,S2: int,P3: int,R2: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S2
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S2 ) ) )
=> ( ( times_times @ int @ P3 @ S2 )
= ( times_times @ int @ R2 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_4006_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
@ ^ [B8: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B8 ) @ F5 ) ) ) ) ).
% su_rel_fun.f_def
thf(fact_4007_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B] :
( ! [A11: A,B7: B,B9: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A11 @ B7 ) @ F5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A11 @ B9 ) @ F5 )
=> ( B7 = B9 ) ) )
=> ( ! [A11: A,P4: $o] :
( ! [B17: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A11 @ B17 ) @ F5 )
=> P4 )
=> P4 )
=> ( ! [A11: A] :
( ( F2 @ A11 )
= ( the @ B
@ ^ [B8: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A11 @ B8 ) @ F5 ) ) )
=> ( su_rel_fun @ A @ B @ F5 @ F2 ) ) ) ) ).
% su_rel_fun.intro
thf(fact_4008_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_4009_set__mset__eq__empty__iff,axiom,
! [A: $tType,M2: multiset @ A] :
( ( ( set_mset @ A @ M2 )
= ( bot_bot @ ( set @ A ) ) )
= ( M2
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% set_mset_eq_empty_iff
thf(fact_4010_set__mset__add__mset__insert,axiom,
! [A: $tType,A4: A,A3: multiset @ A] :
( ( set_mset @ A @ ( add_mset @ A @ A4 @ A3 ) )
= ( insert2 @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ).
% set_mset_add_mset_insert
thf(fact_4011_mset__diff__cancel1elem,axiom,
! [A: $tType,A4: A,B5: multiset @ A] :
( ~ ( member @ A @ A4 @ ( set_mset @ A @ B5 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ B5 )
= ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% mset_diff_cancel1elem
thf(fact_4012_mset__un__cases,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ A3 ) )
=> ( member @ A @ A4 @ ( set_mset @ A @ B5 ) ) ) ) ).
% mset_un_cases
thf(fact_4013_mset__left__cancel__union,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ A3 ) )
=> ( member @ A @ A4 @ ( set_mset @ A @ B5 ) ) ) ) ).
% mset_left_cancel_union
thf(fact_4014_mset__right__cancel__union,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ B5 ) )
=> ( member @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ) ).
% mset_right_cancel_union
thf(fact_4015_ex__Melem__conv,axiom,
! [A: $tType,A3: multiset @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ ( set_mset @ A @ A3 ) ) )
= ( A3
!= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% ex_Melem_conv
thf(fact_4016_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 )
=> ~ ! [B7: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ F5 ) ) ).
% su_rel_fun.surjective
thf(fact_4017_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A,B5: B,B12: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ F5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B12 ) @ F5 )
=> ( B5 = B12 ) ) ) ) ).
% su_rel_fun.unique
thf(fact_4018_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A,B5: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ F5 )
=> ( B5
= ( F2 @ A3 ) ) ) ) ).
% su_rel_fun.repr2
thf(fact_4019_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_4020_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F5: set @ ( product_prod @ A @ B ),F2: A > B,A3: A,B5: B] :
( ( su_rel_fun @ A @ B @ F5 @ F2 )
=> ( ( ( F2 @ A3 )
= B5 )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ F5 ) ) ) ).
% su_rel_fun.repr
thf(fact_4021_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_4022_mset__left__cancel__elem,axiom,
! [A: $tType,A4: A,B3: A,A3: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 ) ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ) ).
% mset_left_cancel_elem
thf(fact_4023_mset__right__cancel__elem,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B3: A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ) ).
% mset_right_cancel_elem
thf(fact_4024_infinite__set__mset__mset__set,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( set_mset @ A @ ( mset_set @ A @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_set_mset_mset_set
thf(fact_4025_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_4026_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_4027_mset__contains__eq,axiom,
! [A: $tType,M: A,M2: multiset @ A] :
( ( member @ A @ M @ ( set_mset @ A @ M2 ) )
= ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M2 @ ( add_mset @ A @ M @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M2 ) ) ).
% mset_contains_eq
thf(fact_4028_mset__union__diff__comm,axiom,
! [A: $tType,T5: A,S: multiset @ A,T3: multiset @ A] :
( ( member @ A @ T5 @ ( set_mset @ A @ S ) )
=> ( ( plus_plus @ ( multiset @ A ) @ T3 @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T5 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T3 @ S ) @ ( add_mset @ A @ T5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_union_diff_comm
thf(fact_4029_diff__union__single__conv2,axiom,
! [A: $tType,A4: A,J: multiset @ A,I: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ J ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J @ I ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I ) ) ) ).
% diff_union_single_conv2
thf(fact_4030_mset__un__single__un__cases,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( ( add_mset @ A @ A4 @ A3 )
= ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) )
=> ( ( ( member @ A @ A4 @ ( set_mset @ A @ B5 ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C6 ) ) )
=> ~ ( ( member @ A @ A4 @ ( set_mset @ A @ C6 ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_un_single_un_cases
thf(fact_4031_su__rel__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( su_rel_fun @ A @ B )
= ( ^ [F7: set @ ( product_prod @ A @ B ),F4: A > B] :
( ! [A5: A,B8: B,B18: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B8 ) @ F7 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B18 ) @ F7 )
=> ( B8 = B18 ) ) )
& ! [A5: A,P2: $o] :
( ! [B8: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B8 ) @ F7 )
=> P2 )
=> P2 )
& ! [A5: A] :
( ( F4 @ A5 )
= ( the @ B
@ ^ [B8: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B8 ) @ F7 ) ) ) ) ) ) ).
% su_rel_fun_def
thf(fact_4032_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N7: multiset @ A,M5: multiset @ A] :
? [A8: A,M0: multiset @ A,K6: multiset @ A] :
( ( M5
= ( add_mset @ A @ A8 @ M0 ) )
& ( N7
= ( plus_plus @ ( multiset @ A ) @ M0 @ K6 ) )
& ! [B6: A] :
( ( member @ A @ B6 @ ( set_mset @ A @ K6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A8 ) @ R5 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_4033_rat__minus__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ B6 @ C5 ) ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_minus_code
thf(fact_4034_rat__plus__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ B6 @ C5 ) ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_plus_code
thf(fact_4035_rat__times__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A8 @ B6 ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_times_code
thf(fact_4036_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P5: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A8: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B6: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ C5 @ B6 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_4037_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P5: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A8: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B6: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ C5 @ B6 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_4038_mult1E,axiom,
! [A: $tType,N: multiset @ A,M2: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M2 ) @ ( mult1 @ A @ R2 ) )
=> ~ ! [A6: A,M02: multiset @ A] :
( ( M2
= ( add_mset @ A @ A6 @ M02 ) )
=> ! [K7: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ M02 @ K7 ) )
=> ~ ! [B15: A] :
( ( member @ A @ B15 @ ( set_mset @ A @ K7 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B15 @ A6 ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_4039_mult1I,axiom,
! [A: $tType,M2: multiset @ A,A4: A,M03: multiset @ A,N: multiset @ A,K2: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( M2
= ( add_mset @ A @ A4 @ M03 ) )
=> ( ( N
= ( plus_plus @ ( multiset @ A ) @ M03 @ K2 ) )
=> ( ! [B2: A] :
( ( member @ A @ B2 @ ( set_mset @ A @ K2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A4 ) @ R2 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M2 ) @ ( mult1 @ A @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_4040_less__add,axiom,
! [A: $tType,N: multiset @ A,A4: A,M03: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ ( add_mset @ A @ A4 @ M03 ) ) @ ( mult1 @ A @ R2 ) )
=> ( ? [M6: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M6 @ M03 ) @ ( mult1 @ A @ R2 ) )
& ( N
= ( add_mset @ A @ A4 @ M6 ) ) )
| ? [K7: multiset @ A] :
( ! [B15: A] :
( ( member @ A @ B15 @ ( set_mset @ A @ K7 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B15 @ A4 ) @ R2 ) )
& ( N
= ( plus_plus @ ( multiset @ A ) @ M03 @ K7 ) ) ) ) ) ).
% less_add
thf(fact_4041_rat__divide__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A8 @ D5 ) @ ( times_times @ int @ C5 @ B6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_divide_code
thf(fact_4042_rat__inverse__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A8: int,B6: int] :
( if @ ( product_prod @ int @ int )
@ ( A8
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A8 ) @ B6 ) @ ( abs_abs @ int @ A8 ) ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_inverse_code
thf(fact_4043_mult__implies__one__step,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),M2: multiset @ A,N: multiset @ A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M2 @ N ) @ ( mult @ A @ R2 ) )
=> ? [I7: multiset @ A,J6: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ I7 @ J6 ) )
& ? [K7: multiset @ A] :
( ( M2
= ( plus_plus @ ( multiset @ A ) @ I7 @ K7 ) )
& ( J6
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_mset @ A @ K7 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_4044_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N2 @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N2 @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_4045_size__diff__se,axiom,
! [A: $tType,T5: A,S: multiset @ A] :
( ( member @ A @ T5 @ ( set_mset @ A @ S ) )
=> ( ( size_size @ ( multiset @ A ) @ S )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_4046_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_4047_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_4048_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_4049_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_4050_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_4051_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_4052_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_4053_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_4054_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_4055_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_4056_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A8: A,B6: A] : ( times_times @ A @ A8 @ ( inverse_inverse @ A @ B6 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_4057_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A8: A,B6: A] : ( times_times @ A @ A8 @ ( inverse_inverse @ A @ B6 ) ) ) ) ) ).
% divide_inverse
thf(fact_4058_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A8: A,B6: A] : ( times_times @ A @ ( inverse_inverse @ A @ B6 ) @ A8 ) ) ) ) ).
% divide_inverse_commute
thf(fact_4059_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_4060_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N2 ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_4061_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_4062_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_4063_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_4064_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_4065_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_4066_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_4067_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_4068_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_4069_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_4070_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_4071_divide__rat__def,axiom,
( ( divide_divide @ rat )
= ( ^ [Q6: rat,R5: rat] : ( times_times @ rat @ Q6 @ ( inverse_inverse @ rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_4072_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_4073_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_4074_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_4075_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_4076_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_4077_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_4078_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N2: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4079_mset__size__le1__cases,axiom,
! [A: $tType,M2: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M2 ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M2
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M3: A] :
( M2
!= ( add_mset @ A @ M3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_4080_mset__size1elem,axiom,
! [A: $tType,P: multiset @ A,Q4: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
=> ( ( member @ A @ Q4 @ ( set_mset @ A @ P ) )
=> ( P
= ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size1elem
thf(fact_4081_one__step__implies__mult,axiom,
! [A: $tType,J: multiset @ A,K2: multiset @ A,R2: set @ ( product_prod @ A @ A ),I: multiset @ A] :
( ( J
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K2 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R2 ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I @ J ) ) @ ( mult @ A @ R2 ) ) ) ) ).
% one_step_implies_mult
thf(fact_4082_multp__code__iff__mult,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),P: A > A > $o,N: multiset @ A,M2: multiset @ A] :
( ( irrefl @ A @ R4 )
=> ( ( trans @ A @ R4 )
=> ( ! [X3: A,Y2: A] :
( ( P @ X3 @ Y2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R4 ) )
=> ( ( multp_code @ A @ P @ N @ M2 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M2 ) @ ( mult @ A @ R4 ) ) ) ) ) ) ).
% multp_code_iff_mult
thf(fact_4083_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),P: A > A > $o,N: multiset @ A,M2: multiset @ A] :
( ( irrefl @ A @ R4 )
=> ( ( trans @ A @ R4 )
=> ( ! [X3: A,Y2: A] :
( ( P @ X3 @ Y2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R4 ) )
=> ( ( multeqp_code @ A @ P @ N @ M2 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R4 ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
thf(fact_4084_mset__size2elem,axiom,
! [A: $tType,P: multiset @ A,Q4: A,Q5: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q5 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
=> ( P
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_4085_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_4086_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_4087_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,N: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( add_mset @ A @ X @ N ) )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ N ) ) ) ) ).
% prod_mset.add_mset
thf(fact_4088_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ A,B5: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A3 ) @ ( comm_m9189036328036947845d_mset @ A @ B5 ) ) ) ) ).
% prod_mset_Un
thf(fact_4089_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: multiset @ A,N: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M2 @ N ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M2 ) @ ( comm_m9189036328036947845d_mset @ A @ N ) ) ) ) ).
% prod_mset.union
thf(fact_4090_subset__mset_Ofinite__has__minimal,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( 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_4091_subset__mset_Ofinite__has__maximal,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( 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_4092_mset__le__incr__right2,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A4 @ B3 )
=> ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B3 ) ) ) ).
% mset_le_incr_right2
thf(fact_4093_mset__le__incr__right1,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A4 @ B3 )
=> ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B3 @ C2 ) ) ) ).
% mset_le_incr_right1
thf(fact_4094_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset @ A,A4: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A4 ) @ B3 )
=> ( subseteq_mset @ A @ A4 @ B3 ) ) ).
% mset_le_decr_left2
thf(fact_4095_mset__le__decr__left1,axiom,
! [A: $tType,A4: multiset @ A,C2: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C2 ) @ B3 )
=> ( subseteq_mset @ A @ A4 @ B3 ) ) ).
% mset_le_decr_left1
thf(fact_4096_mset__union__subset,axiom,
! [A: $tType,A3: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) @ C6 )
=> ( ( subseteq_mset @ A @ A3 @ C6 )
& ( subseteq_mset @ A @ B5 @ C6 ) ) ) ).
% mset_union_subset
thf(fact_4097_mset__le__distrib,axiom,
! [A: $tType,X4: multiset @ A,A3: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ X4 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) )
=> ~ ! [Xa4: multiset @ A,Xb3: multiset @ A] :
( ( X4
= ( plus_plus @ ( multiset @ A ) @ Xa4 @ Xb3 ) )
=> ( ( subseteq_mset @ A @ Xa4 @ A3 )
=> ~ ( subseteq_mset @ A @ Xb3 @ B5 ) ) ) ) ).
% mset_le_distrib
thf(fact_4098_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys )
=> ~ ! [Zs: multiset @ A] :
( Ys
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs ) ) ) ).
% mset_le_addE
thf(fact_4099_mset__le__subtract,axiom,
! [A: $tType,A3: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B5 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A3 @ C6 ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ).
% mset_le_subtract
thf(fact_4100_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A4: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A4 ) @ B3 )
=> ( subseteq_mset @ A @ A4 @ B3 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_4101_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A4: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A4 ) @ B3 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B3 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_4102_mset__le__single__cases,axiom,
! [A: $tType,M2: multiset @ A,A4: A] :
( ( subseteq_mset @ A @ M2 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M2
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M2
= ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_4103_mset__le__add__mset,axiom,
! [A: $tType,X: A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B5 ) @ C6 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C6 )
& ( subseteq_mset @ A @ B5 @ C6 ) ) ) ).
% mset_le_add_mset
thf(fact_4104_mset__le__subtract__left,axiom,
! [A: $tType,A3: multiset @ A,B5: multiset @ A,X4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) @ X4 )
=> ( ( subseteq_mset @ A @ B5 @ ( minus_minus @ ( multiset @ A ) @ X4 @ A3 ) )
& ( subseteq_mset @ A @ A3 @ X4 ) ) ) ).
% mset_le_subtract_left
thf(fact_4105_mset__le__subtract__right,axiom,
! [A: $tType,A3: multiset @ A,B5: multiset @ A,X4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) @ X4 )
=> ( ( subseteq_mset @ A @ A3 @ ( minus_minus @ ( multiset @ A ) @ X4 @ B5 ) )
& ( subseteq_mset @ A @ B5 @ X4 ) ) ) ).
% mset_le_subtract_right
thf(fact_4106_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_4107_subset__mset_OcInf__greatest,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X4 )
=> ( subseteq_mset @ A @ Z2 @ X3 ) )
=> ( subseteq_mset @ A @ Z2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X4 ) ) ) ) ).
% subset_mset.cInf_greatest
thf(fact_4108_subset__mset_OcInf__eq__non__empty,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X4 )
=> ( subseteq_mset @ A @ A4 @ X3 ) )
=> ( ! [Y2: multiset @ A] :
( ! [X6: multiset @ A] :
( ( member @ ( multiset @ A ) @ X6 @ X4 )
=> ( subseteq_mset @ A @ Y2 @ X6 ) )
=> ( subseteq_mset @ A @ Y2 @ A4 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X4 )
= A4 ) ) ) ) ).
% subset_mset.cInf_eq_non_empty
thf(fact_4109_subset__mset_OcSup__least,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X4 )
=> ( subseteq_mset @ A @ X3 @ Z2 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ X4 ) @ Z2 ) ) ) ).
% subset_mset.cSup_least
thf(fact_4110_subset__mset_OcSup__eq__non__empty,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X4 )
=> ( subseteq_mset @ A @ X3 @ A4 ) )
=> ( ! [Y2: multiset @ A] :
( ! [X6: multiset @ A] :
( ( member @ ( multiset @ A ) @ X6 @ X4 )
=> ( subseteq_mset @ A @ X6 @ Y2 ) )
=> ( subseteq_mset @ A @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X4 )
= A4 ) ) ) ) ).
% subset_mset.cSup_eq_non_empty
thf(fact_4111_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B5: multiset @ A,X4: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B5 ) @ X4 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X4 @ B5 ) )
& ( subseteq_mset @ A @ B5 @ X4 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_4112_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B5: multiset @ A,X4: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B5 ) @ X4 )
=> ( ( subseteq_mset @ A @ B5 @ ( minus_minus @ ( multiset @ A ) @ X4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X4 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_4113_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_4114_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A3: set @ B,M: multiset @ A,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subseteq_mset @ A @ M @ ( F2 @ X3 ) ) )
=> ( subseteq_mset @ A @ M @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_4115_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A3: set @ B,F2: B > ( multiset @ A ),M2: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subseteq_mset @ A @ ( F2 @ X3 ) @ M2 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ M2 ) ) ) ).
% subset_mset.cSUP_least
thf(fact_4116_mset__le__mono__add__single,axiom,
! [A: $tType,A4: A,Ys: multiset @ A,B3: A,Ws: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ Ys ) )
=> ( ( member @ A @ B3 @ ( set_mset @ A @ Ws ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys @ Ws ) ) ) ) ).
% mset_le_mono_add_single
thf(fact_4117_mset__union__subset__s,axiom,
! [A: $tType,A4: A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ B5 ) @ C6 )
=> ( ( member @ A @ A4 @ ( set_mset @ A @ C6 ) )
& ( subseteq_mset @ A @ B5 @ C6 ) ) ) ).
% mset_union_subset_s
thf(fact_4118_mset__2dist2__cases,axiom,
! [A: $tType,A4: A,B3: A,A3: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A3 )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B5 )
=> ( ( ( member @ A @ A4 @ ( set_mset @ A @ A3 ) )
=> ~ ( member @ A @ B3 @ ( set_mset @ A @ B5 ) ) )
=> ~ ( ( member @ A @ A4 @ ( set_mset @ A @ B5 ) )
=> ~ ( member @ A @ B3 @ ( set_mset @ A @ A3 ) ) ) ) ) ) ) ).
% mset_2dist2_cases
thf(fact_4119_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_4120_subset__mset_OatLeastAtMost__singleton__iff,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,C2: multiset @ A] :
( ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A4 @ B3 )
= ( insert2 @ ( multiset @ A ) @ C2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( ( A4 = B3 )
& ( B3 = C2 ) ) ) ).
% subset_mset.atLeastAtMost_singleton_iff
thf(fact_4121_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,G: B > ( multiset @ A ),B5: set @ B,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B5 )
=> ( subseteq_mset @ A @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.cINF_superset_mono
thf(fact_4122_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,G: B > ( multiset @ A ),B5: set @ B,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subseteq_mset @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B5 ) ) ) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
thf(fact_4123_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_4124_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_4125_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_4126_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_4127_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_4128_subset__mset_OcSup__mono,axiom,
! [A: $tType,B5: set @ ( multiset @ A ),A3: set @ ( multiset @ A )] :
( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ! [B2: multiset @ A] :
( ( member @ ( multiset @ A ) @ B2 @ B5 )
=> ? [X6: multiset @ A] :
( ( member @ ( multiset @ A ) @ X6 @ A3 )
& ( subseteq_mset @ A @ B2 @ X6 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ B5 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.cSup_mono
thf(fact_4129_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_4130_subset__mset_OcSup__subset__mono,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B5 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B5 )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B5 ) ) ) ) ) ).
% subset_mset.cSup_subset_mono
thf(fact_4131_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_4132_subset__mset_OcInf__mono,axiom,
! [A: $tType,B5: set @ ( multiset @ A ),A3: set @ ( multiset @ A )] :
( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ! [B2: multiset @ A] :
( ( member @ ( multiset @ A ) @ B2 @ B5 )
=> ? [X6: multiset @ A] :
( ( member @ ( multiset @ A ) @ X6 @ A3 )
& ( subseteq_mset @ A @ X6 @ B2 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B5 ) ) ) ) ) ).
% subset_mset.cInf_mono
thf(fact_4133_subset__mset_OcInf__superset__mono,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B5 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B5 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ B5 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.cInf_superset_mono
thf(fact_4134_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_4135_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ B,G: C > ( multiset @ A ),B5: set @ C,F2: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G @ B5 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A3 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B5 )
& ( subseteq_mset @ A @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G @ B5 ) ) ) ) ) ) ).
% subset_mset.cSUP_mono
thf(fact_4136_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_4137_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B5: set @ B,F2: C > ( multiset @ A ),A3: set @ C,G: B > ( multiset @ A )] :
( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B5 )
=> ? [X6: C] :
( ( member @ C @ X6 @ A3 )
& ( subseteq_mset @ A @ ( F2 @ X6 ) @ ( G @ M3 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B5 ) ) ) ) ) ) ).
% subset_mset.cINF_mono
thf(fact_4138_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_4139_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] :
! [Y3: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y3 @ S )
=> ( subseteq_mset @ A @ Y3 @ X2 ) ) ) ) ) ) ) ).
% subset_mset.cSup_cInf
thf(fact_4140_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] :
! [Y3: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y3 @ S )
=> ( subseteq_mset @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% subset_mset.cInf_cSup
thf(fact_4141_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A3: C > ( multiset @ A ),I: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I ) )
=> ( ( I
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_4142_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: ( multiset @ A ) > B,A3: C > ( multiset @ A ),I: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F2 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I ) )
=> ( ( I
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( F2 @ ( A3 @ X2 ) )
@ I ) )
@ ( F2 @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_4143_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_4144_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_4145_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_4146_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_finite2 @ 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_4147_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_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ord_less @ nat @ ( zero_zero @ nat )
@ ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [M5: multiset @ A] : ( count @ A @ M5 @ I4 )
@ A3 ) ) ) ) ) ) ) ).
% Sup_multiset_in_multiset
thf(fact_4148_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),B5: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B5 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B5 ) ) ) ) ) ) ) ) ).
% subset_mset.cINF_union
thf(fact_4149_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_4150_mset__empty__count,axiom,
! [A: $tType,M2: multiset @ A] :
( ( ! [P5: A] :
( ( count @ A @ M2 @ P5 )
= ( zero_zero @ nat ) ) )
= ( M2
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% mset_empty_count
thf(fact_4151_count__ne__remove,axiom,
! [A: $tType,X: A,T5: A,S: multiset @ A] :
( ( X != T5 )
=> ( ( count @ A @ S @ X )
= ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T5 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X ) ) ) ).
% count_ne_remove
thf(fact_4152_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X: set @ ( multiset @ A )] :
( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ X ) )
= ( ^ [I4: 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
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X ) ) ) ) ) ) ).
% Inf_multiset.rep_eq
thf(fact_4153_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_4154_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set @ A,A4: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( member @ A @ A4 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A4 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A4 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A4 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_4155_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B5 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B5 ) ) ) ) ) ) ).
% subset_mset.less_eq_cInf_inter
thf(fact_4156_subset__mset_OcInf__union__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B5 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B5 ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B5 ) ) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
thf(fact_4157_subset__mset_OcInf__insert,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X4 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X4 ) )
= ( inter_mset @ A @ A4 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X4 ) ) ) ) ) ).
% subset_mset.cInf_insert
thf(fact_4158_subset__mset_OcInf__insert__If,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),A4: multiset @ A] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X4 )
=> ( ( ( X4
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X4 ) )
= A4 ) )
& ( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X4 ) )
= ( inter_mset @ A @ A4 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X4 ) ) ) ) ) ) ).
% subset_mset.cInf_insert_If
thf(fact_4159_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),G: 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 ) @ G @ A3 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A3 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A8: B] : ( inter_mset @ A @ ( F2 @ A8 ) @ ( G @ A8 ) )
@ A3 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_4160_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_4161_Sup__multiset__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) )
= ( ^ [A5: set @ ( multiset @ A )] :
( if @ ( multiset @ A )
@ ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 ) )
@ ( abs_multiset @ A
@ ^ [I4: A] :
( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X7: multiset @ A] : ( count @ A @ X7 @ I4 )
@ A5 ) ) )
@ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Sup_multiset_def
thf(fact_4162_size__multiset__eq,axiom,
! [A: $tType] :
( ( size_multiset @ A )
= ( ^ [F4: A > nat,M5: multiset @ A] :
( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count @ A @ M5 @ X2 ) @ ( suc @ ( F4 @ X2 ) ) )
@ ( set_mset @ A @ M5 ) ) ) ) ).
% size_multiset_eq
thf(fact_4163_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_4164_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),B5: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B5 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ B5 ) ) ) ) ) ) ) ) ).
% subset_mset.cSUP_union
thf(fact_4165_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_4166_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,X: B,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X @ A3 ) ) )
= ( times_times @ A @ ( G @ X ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A3 ) ) ) ) ) ).
% prod_mset.insert
thf(fact_4167_mset__map__id,axiom,
! [B: $tType,A: $tType,F2: B > A,G: A > B,X4: multiset @ A] :
( ! [X3: A] :
( ( F2 @ ( G @ X3 ) )
= X3 )
=> ( ( image_mset @ B @ A @ F2 @ ( image_mset @ A @ B @ G @ X4 ) )
= X4 ) ) ).
% mset_map_id
thf(fact_4168_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F2: B > A,P: multiset @ B,M1: multiset @ A,M22: multiset @ A] :
( ( ( image_mset @ B @ A @ F2 @ P )
= ( plus_plus @ ( multiset @ A ) @ M1 @ M22 ) )
=> ~ ! [P12: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P22 ) )
=> ( ( ( image_mset @ B @ A @ F2 @ P12 )
= M1 )
=> ( ( image_mset @ B @ A @ F2 @ P22 )
!= M22 ) ) ) ) ).
% mset_map_split_orig
thf(fact_4169_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ B,B5: multiset @ B,G: B > A] :
( ( ( inter_mset @ B @ A3 @ B5 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A3 @ B5 ) ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A3 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod_mset.union_disjoint
thf(fact_4170_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F2: B > A,P: multiset @ B,M1: multiset @ A,M22: multiset @ A] :
( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F2 @ P ) @ ( plus_plus @ ( multiset @ A ) @ M1 @ M22 ) )
=> ~ ! [P12: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P22 ) )
=> ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F2 @ P12 ) @ M1 )
=> ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F2 @ P22 ) @ M22 ) ) ) ) ).
% mset_map_split_orig_le
thf(fact_4171_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M2: multiset @ ( product_prod @ A @ B ),F2: A > B > C,G: A > B > C] :
( ! [X3: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M2 ) )
=> ( ( F2 @ X3 @ Y2 )
= ( G @ X3 @ Y2 ) ) )
=> ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ M2 )
= ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G ) @ M2 ) ) ) ).
% image_mset_cong_pair
thf(fact_4172_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A3 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A3 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H2 @ A3 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_4173_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( X2 = Y ) @ C2 @ ( one_one @ A ) )
@ A3 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A3 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_4174_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( Y = X2 ) @ C2 @ ( one_one @ A ) )
@ A3 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A3 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_4175_subset__mset_OcSup__union__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B5 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B5 ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B5 ) ) ) ) ) ) ) ).
% subset_mset.cSup_union_distrib
thf(fact_4176_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B5 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A3 @ B5 ) ) @ ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B5 ) ) ) ) ) ) ).
% subset_mset.cSup_inter_less_eq
thf(fact_4177_subset__mset_OcSup__insert,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),A4: multiset @ A] :
( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X4 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X4 ) )
= ( union_mset @ A @ A4 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X4 ) ) ) ) ) ).
% subset_mset.cSup_insert
thf(fact_4178_subset__mset_OcSup__insert__If,axiom,
! [A: $tType,X4: set @ ( multiset @ A ),A4: multiset @ A] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X4 )
=> ( ( ( X4
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X4 ) )
= A4 ) )
& ( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert2 @ ( multiset @ A ) @ A4 @ X4 ) )
= ( union_mset @ A @ A4 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X4 ) ) ) ) ) ) ).
% subset_mset.cSup_insert_If
thf(fact_4179_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),G: 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 ) @ G @ A3 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A3 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A8: B] : ( union_mset @ A @ ( F2 @ A8 ) @ ( G @ A8 ) )
@ A3 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_4180_type__definition__multiset,axiom,
! [A: $tType] :
( type_definition @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ ( abs_multiset @ A )
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) ) ) ) ).
% type_definition_multiset
thf(fact_4181_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_4182_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 )
@ ^ [A5: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A5
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A5 ) ) ) ) ) ).
% Inf_multiset_def
thf(fact_4183_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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] :
? [A8: multiset @ A] :
( ( Uu
= ( inter_mset @ A @ X @ A8 ) )
& ( member @ ( multiset @ A ) @ A8 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup1_distrib
thf(fact_4184_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B5 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A8: multiset @ A,B6: multiset @ A] :
( ( Uu
= ( inter_mset @ A @ A8 @ B6 ) )
& ( member @ ( multiset @ A ) @ A8 @ A3 )
& ( member @ ( multiset @ A ) @ B6 @ B5 ) ) ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
thf(fact_4185_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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] :
? [A8: multiset @ A] :
( ( Uu
= ( union_mset @ A @ X @ A8 ) )
& ( member @ ( multiset @ A ) @ A8 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf1_distrib
thf(fact_4186_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_4187_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_4188_subset__mset_OInf__fin_Oinsert,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4189_subset__mset_OSup__fin_Oinsert,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4190_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( 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_4191_subset__mset_OInf__fin_Ohom__commute,axiom,
! [A: $tType,H2: ( multiset @ A ) > ( multiset @ A ),N: set @ ( multiset @ A )] :
( ! [X3: multiset @ A,Y2: multiset @ A] :
( ( H2 @ ( inter_mset @ A @ X3 @ Y2 ) )
= ( inter_mset @ A @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ N )
=> ( ( N
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H2 @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ N ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H2 @ N ) ) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
thf(fact_4192_subset__mset_OSup__fin_Ohom__commute,axiom,
! [A: $tType,H2: ( multiset @ A ) > ( multiset @ A ),N: set @ ( multiset @ A )] :
( ! [X3: multiset @ A,Y2: multiset @ A] :
( ( H2 @ ( union_mset @ A @ X3 @ Y2 ) )
= ( union_mset @ A @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ N )
=> ( ( N
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H2 @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ N ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H2 @ N ) ) ) ) ) ) ).
% subset_mset.Sup_fin.hom_commute
thf(fact_4193_subset__mset_OInf__fin_OboundedE,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) )
=> ! [A12: multiset @ A] :
( ( member @ ( multiset @ A ) @ A12 @ A3 )
=> ( subseteq_mset @ A @ X @ A12 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
thf(fact_4194_subset__mset_OInf__fin_OboundedI,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4195_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4196_subset__mset_OSup__fin_OboundedE,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ X )
=> ! [A12: multiset @ A] :
( ( member @ ( multiset @ A ) @ A12 @ A3 )
=> ( subseteq_mset @ A @ A12 @ X ) ) ) ) ) ).
% subset_mset.Sup_fin.boundedE
thf(fact_4197_subset__mset_OSup__fin_OboundedI,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4198_subset__mset_OSup__fin_Obounded__iff,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4199_subset__mset_OcInf__eq__Inf__fin,axiom,
! [A: $tType,X4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X4 )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ X4 ) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
thf(fact_4200_subset__mset_OcSup__eq__Sup__fin,axiom,
! [A: $tType,X4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ X4 )
=> ( ( X4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X4 )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ X4 ) ) ) ) ).
% subset_mset.cSup_eq_Sup_fin
thf(fact_4201_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4202_subset__mset_OInf__fin_Oremove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4203_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4204_subset__mset_OInf__fin_Oclosed,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A,Y2: multiset @ A] : ( member @ ( multiset @ A ) @ ( inter_mset @ A @ X3 @ Y2 ) @ ( insert2 @ ( multiset @ A ) @ X3 @ ( insert2 @ ( multiset @ A ) @ Y2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ A3 ) ) ) ) ).
% subset_mset.Inf_fin.closed
thf(fact_4205_subset__mset_OSup__fin_Oinsert__remove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4206_subset__mset_OSup__fin_Oremove,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4207_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( 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_4208_subset__mset_OSup__fin_Oclosed,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A,Y2: multiset @ A] : ( member @ ( multiset @ A ) @ ( union_mset @ A @ X3 @ Y2 ) @ ( insert2 @ ( multiset @ A ) @ X3 @ ( insert2 @ ( multiset @ A ) @ Y2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ A3 ) ) ) ) ).
% subset_mset.Sup_fin.closed
thf(fact_4209_subset__mset_OInf__fin_Osubset,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B5 @ A3 )
=> ( ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B5 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset
thf(fact_4210_subset__mset_OSup__fin_Osubset,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B5 @ A3 )
=> ( ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B5 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset
thf(fact_4211_subset__mset_OInf__fin_Ounion,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B5 ) )
= ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B5 ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
thf(fact_4212_subset__mset_OSup__fin_Ounion,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A3 @ B5 ) )
= ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B5 ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.union
thf(fact_4213_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B5 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
thf(fact_4214_subset__mset_OSup__fin_Osubset__imp,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A3 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B5 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset_imp
thf(fact_4215_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,A3: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A3 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B5 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A8: multiset @ A,B6: multiset @ A] :
( ( Uu
= ( union_mset @ A @ A8 @ B6 ) )
& ( member @ ( multiset @ A ) @ A8 @ A3 )
& ( member @ ( multiset @ A ) @ B6 @ B5 ) ) ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
thf(fact_4216_multiset_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R3: A > B > $o,A8: multiset @ A,B6: multiset @ B] :
? [Z3: multiset @ ( product_prod @ A @ B )] :
( ( member @ ( multiset @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X2: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ) )
& ( ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A8 )
& ( ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B6 ) ) ) ) ).
% multiset.in_rel
thf(fact_4217_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_4218_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A3: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( X2 = Y ) @ C2 @ ( zero_zero @ A ) )
@ A3 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A3 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_4219_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A3: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( if @ A @ ( Y = X2 ) @ C2 @ ( zero_zero @ A ) )
@ A3 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A3 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_4220_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat,A4: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N2 @ A4 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ A4 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_4221_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,G: C > B,B5: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F2 @ A3 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G @ B5 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I4: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I4 ) @ ( G @ J3 ) )
@ B5 ) )
@ A3 ) ) ) ) ).
% sum_mset_product
thf(fact_4222_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,M2: multiset @ B,C2: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F2 @ M2 ) ) @ C2 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ M2 ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_4223_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F2: B > A,M2: multiset @ B] :
( ( times_times @ A @ C2 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F2 @ M2 ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ M2 ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_4224_wcount__def,axiom,
! [A: $tType] :
( ( wcount @ A )
= ( ^ [F4: A > nat,M5: multiset @ A,X2: A] : ( times_times @ nat @ ( count @ A @ M5 @ X2 ) @ ( suc @ ( F4 @ X2 ) ) ) ) ) ).
% wcount_def
thf(fact_4225_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: B] : ( member @ B @ X2 @ ( range2 @ A @ B @ R2 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_4226_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% Range_def
thf(fact_4227_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B6: A] :
( ( B6 != A8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_4228_RangepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B3: B] :
( ( rangep @ A @ B @ R2 @ B3 )
=> ~ ! [A6: A] :
~ ( R2 @ A6 @ B3 ) ) ).
% RangepE
thf(fact_4229_RangePI,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: A,B3: B] :
( ( R2 @ A4 @ B3 )
=> ( rangep @ A @ B @ R2 @ B3 ) ) ).
% RangePI
thf(fact_4230_Rangep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rangep @ A @ B )
= ( ^ [R5: A > B > $o,A8: B] :
? [B6: A,C5: B] :
( ( A8 = C5 )
& ( R5 @ B6 @ C5 ) ) ) ) ).
% Rangep.simps
thf(fact_4231_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: B] :
( ( rangep @ A @ B @ R2 @ A4 )
=> ~ ! [A6: A] :
~ ( R2 @ A6 @ A4 ) ) ).
% Rangep.cases
thf(fact_4232_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A5: set @ A,R3: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A15: A,A24: A] :
( ( Uu
= ( product_Pair @ A @ A @ A15 @ A24 ) )
& ( member @ A @ A15 @ A5 )
& ( member @ A @ A24 @ A5 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ A15 ) @ ( F4 @ A24 ) ) @ R3 ) ) ) ) ) ).
% relInvImage_def
thf(fact_4233_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X2: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X2 ) ) @ ( product_snd @ B @ C @ ( F4 @ X2 ) ) ) ) ) ).
% scomp_unfold
thf(fact_4234_antisymp__antisym__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisymp_antisym_eq
thf(fact_4235_sub__BitM__One__eq,axiom,
! [N2: num] :
( ( neg_numeral_sub @ int @ ( bitM @ N2 ) @ one2 )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N2 @ one2 ) ) ) ).
% sub_BitM_One_eq
thf(fact_4236_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X2: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X2 ) ) ) ) ).
% scomp_apply
thf(fact_4237_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F2: A > ( product_prod @ E @ F3 ),G: E > F3 > ( product_prod @ C @ D ),H2: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F2 @ G ) @ H2 )
= ( product_scomp @ A @ E @ F3 @ B @ F2
@ ^ [X2: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G @ X2 ) @ H2 ) ) ) ).
% scomp_scomp
thf(fact_4238_antisympD,axiom,
! [A: $tType,R2: A > A > $o,A4: A,B3: A] :
( ( antisymp @ A @ R2 )
=> ( ( R2 @ A4 @ B3 )
=> ( ( R2 @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% antisympD
thf(fact_4239_antisympI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A,Y2: A] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ X3 )
=> ( X3 = Y2 ) ) )
=> ( antisymp @ A @ R2 ) ) ).
% antisympI
thf(fact_4240_antisymp__def,axiom,
! [A: $tType] :
( ( antisymp @ A )
= ( ^ [R5: A > A > $o] :
! [X2: A,Y3: A] :
( ( R5 @ X2 @ Y3 )
=> ( ( R5 @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisymp_def
thf(fact_4241_antisymp__equality,axiom,
! [A: $tType] :
( antisymp @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) ).
% antisymp_equality
thf(fact_4242_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_4243_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_4244_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X2: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X2 ) ) ) ) ).
% scomp_def
thf(fact_4245_antisymp__less__eq,axiom,
! [A: $tType,R2: A > A > $o,S2: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R2 @ S2 )
=> ( ( antisymp @ A @ S2 )
=> ( antisymp @ A @ R2 ) ) ) ).
% antisymp_less_eq
thf(fact_4246_antisym__bot,axiom,
! [A: $tType] : ( antisymp @ A @ ( bot_bot @ ( A > A > $o ) ) ) ).
% antisym_bot
thf(fact_4247_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bitM @ N2 ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_4248_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > ( product_prod @ B @ C ),G: A > B,H2: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 )
= G )
=> ( ( H2
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) )
= ( ( bNF_convol @ A @ B @ C @ G @ H2 )
= F2 ) ) ) ).
% convol_expand_snd'
thf(fact_4249_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > ( product_prod @ B @ C ),G: A > B] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 )
= G )
=> ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) )
= F2 ) ) ).
% convol_expand_snd
thf(fact_4250_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > A] :
( ( bind @ B @ A @ A3
@ ^ [X2: B] : ( insert2 @ A @ ( F2 @ X2 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F2 @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_4251_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_4252_empty__bind,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A )] :
( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_4253_Set_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B5: A > ( set @ C ),C6: C > ( set @ B )] :
( ( bind @ C @ B @ ( bind @ A @ C @ A3 @ B5 ) @ C6 )
= ( bind @ A @ B @ A3
@ ^ [X2: A] : ( bind @ C @ B @ ( B5 @ X2 ) @ C6 ) ) ) ).
% Set.bind_bind
thf(fact_4254_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F4: A > B,G4: A > C,A8: A] : ( product_Pair @ B @ C @ ( F4 @ A8 ) @ ( G4 @ A8 ) ) ) ) ).
% convol_def
thf(fact_4255_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F2: C > B,G: C > A,X: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F2 @ G @ X ) )
= ( G @ X ) ) ).
% snd_convol'
thf(fact_4256_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind @ A @ B @ A3
@ ^ [Uu: A] : B5 )
= B5 ) ) ).
% nonempty_bind_const
thf(fact_4257_bind__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,B5: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind @ B @ A @ A3
@ ^ [Uu: B] : B5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind @ B @ A @ A3
@ ^ [Uu: B] : B5 )
= B5 ) ) ) ).
% bind_const
thf(fact_4258_Set_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind @ A @ B )
= ( ^ [A5: set @ A,F4: A > ( set @ B )] :
( collect @ B
@ ^ [X2: B] :
? [Y3: set @ B] :
( ( member @ ( set @ B ) @ Y3 @ ( image2 @ A @ ( set @ B ) @ F4 @ A5 ) )
& ( member @ B @ X2 @ Y3 ) ) ) ) ) ).
% Set.bind_def
thf(fact_4259_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G: A > C] :
( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F2 @ G ) )
= F2 ) ).
% fst_convol
thf(fact_4260_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > C,G: A > B] :
( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F2 @ G ) )
= G ) ).
% snd_convol
thf(fact_4261_convol__image__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G: D > B,R4: A > B > $o] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( bNF_convol @ ( product_prod @ C @ D ) @ A @ B @ ( comp @ C @ A @ ( product_prod @ C @ D ) @ F2 @ ( product_fst @ C @ D ) ) @ ( comp @ D @ B @ ( product_prod @ C @ D ) @ G @ ( product_snd @ C @ D ) ) ) @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F2 @ G @ R4 ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) ).
% convol_image_vimage2p
thf(fact_4262_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_4263_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_4264_Func__empty,axiom,
! [B: $tType,A: $tType,B5: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( insert2 @ ( A > B )
@ ^ [X2: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_4265_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_4266_predicate2D__vimage2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R4: A > B > $o,F2: A > C,G: B > D,S: C > D > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ R4 @ ( bNF_vimage2p @ A @ C @ B @ D @ $o @ F2 @ G @ S ) )
=> ( ( R4 @ X @ Y )
=> ( S @ ( F2 @ X ) @ ( G @ Y ) ) ) ) ).
% predicate2D_vimage2p
thf(fact_4267_vimage2pI,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R4: A > B > $o,F2: C > A,X: C,G: D > B,Y: D] :
( ( R4 @ ( F2 @ X ) @ ( G @ Y ) )
=> ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F2 @ G @ R4 @ X @ Y ) ) ).
% vimage2pI
thf(fact_4268_vimage2p__def,axiom,
! [B: $tType,E: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ D @ B @ E @ C )
= ( ^ [F4: A > D,G4: B > E,R3: D > E > C,X2: A,Y3: B] : ( R3 @ ( F4 @ X2 ) @ ( G4 @ Y3 ) ) ) ) ).
% vimage2p_def
thf(fact_4269_rel__fun__iff__leq__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ B @ C @ D )
= ( ^ [R3: A > B > $o,S6: C > D > $o,F4: A > C,G4: B > D] : ( ord_less_eq @ ( A > B > $o ) @ R3 @ ( bNF_vimage2p @ A @ C @ B @ D @ $o @ F4 @ G4 @ S6 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_4270_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_4271_Abs__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: A > B,Abs1: B > A,Rep22: C > D,Abs22: D > C,R4: B > D > $o] :
( ( type_definition @ A @ B @ Rep1 @ Abs1 @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep22 @ Abs22 @ ( top_top @ ( set @ D ) ) )
=> ( bNF_rel_fun @ B @ D @ A @ C @ R4 @ ( bNF_vimage2p @ A @ B @ C @ D @ $o @ Rep1 @ Rep22 @ R4 ) @ Abs1 @ Abs22 ) ) ) ).
% Abs_transfer
thf(fact_4272_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep2: A > B,Abs2: B > A,G: D > C,P: C > $o,H2: C > A] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ D @ C @ B @ A @ $o @ G @ Abs2 @ ( bNF_Grp @ C @ A @ ( collect @ C @ P ) @ H2 ) )
= ( bNF_Grp @ D @ B
@ ( collect @ D
@ ^ [X2: D] : ( P @ ( G @ X2 ) ) )
@ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep2 @ H2 ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_4273_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep2: A > B,Abs2: B > A,F2: C > D,P: D > $o,H2: D > B] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ D @ A @ B @ $o @ F2 @ Rep2 @ ( bNF_Grp @ D @ B @ ( collect @ D @ P ) @ H2 ) )
= ( bNF_Grp @ C @ A
@ ( collect @ C
@ ^ [X2: C] : ( P @ ( F2 @ X2 ) ) )
@ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs2 @ H2 ) @ F2 ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_4274_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A3: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A3 @ F2 ) ) ) )
=> ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) @ Z2 )
= ( product_snd @ A @ B @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_4275_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_4276_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N2: int] :
( ( power_int @ A @ ( one_one @ A ) @ N2 )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_4277_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W: num,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_4278_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_4279_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_4280_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N2: num] :
( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_4281_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int,B3: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ B3 ) )
= B3 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_4282_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_4283_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N2: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_4284_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N2: num,B3: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) ) @ B3 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) @ B3 ) ) ) ).
% power_int_add_numeral2
thf(fact_4285_GrpI,axiom,
! [B: $tType,A: $tType,F2: B > A,X: B,Y: A,A3: set @ B] :
( ( ( F2 @ X )
= Y )
=> ( ( member @ B @ X @ A3 )
=> ( bNF_Grp @ B @ A @ A3 @ F2 @ X @ Y ) ) ) ).
% GrpI
thf(fact_4286_GrpE,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X: A,Y: B] :
( ( bNF_Grp @ A @ B @ A3 @ F2 @ X @ Y )
=> ~ ( ( ( F2 @ X )
= Y )
=> ~ ( member @ A @ X @ A3 ) ) ) ).
% GrpE
thf(fact_4287_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A5: set @ A,F4: A > B,A8: A,B6: B] :
( ( B6
= ( F4 @ A8 ) )
& ( member @ A @ A8 @ A5 ) ) ) ) ).
% Grp_def
thf(fact_4288_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_4289_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N2 ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_commutes
thf(fact_4290_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N2: int] :
( ( power_int @ A @ X @ ( times_times @ int @ M @ N2 ) )
= ( power_int @ A @ ( power_int @ A @ X @ M ) @ N2 ) ) ) ).
% power_int_mult
thf(fact_4291_rel__filter__Grp,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( rel_filter @ A @ B @ ( bNF_Grp @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
= ( bNF_Grp @ ( filter @ A ) @ ( filter @ B ) @ ( top_top @ ( set @ ( filter @ A ) ) ) @ ( filtermap @ A @ B @ F2 ) ) ) ).
% rel_filter_Grp
thf(fact_4292_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N2 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_one_over
thf(fact_4293_Grp__UNIV__idI,axiom,
! [A: $tType,X: A,Y: A] :
( ( X = Y )
=> ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) @ X @ Y ) ) ).
% Grp_UNIV_idI
thf(fact_4294_eq__alt,axiom,
! [A: $tType] :
( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) ) ) ).
% eq_alt
thf(fact_4295_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A3: set @ A,F2: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A3 @ F2 ) ) ) )
=> ( member @ A @ ( product_fst @ A @ B @ Z2 ) @ A3 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_4296_Grp__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( A > B > $o ) @ ( bNF_Grp @ A @ B @ A3 @ F2 ) @ ( bNF_Grp @ A @ B @ B5 @ F2 ) ) ) ).
% Grp_mono
thf(fact_4297_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( ( M
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( one_one @ A ) ) )
& ( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_4298_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N: int,A4: A] :
( ( ord_less_eq @ int @ N2 @ N )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A4 @ N2 ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ).
% power_int_increasing
thf(fact_4299_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N: int,A4: A] :
( ( ord_less @ int @ N2 @ N )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N2 ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_4300_power__int__minus__one__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) ) ) ).
% power_int_minus_one_minus
thf(fact_4301_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_4302_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N: int,A4: A] :
( ( ord_less @ int @ N2 @ N )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N ) @ ( power_int @ A @ A4 @ N2 ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_4303_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% one_le_power_int
thf(fact_4304_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A4 @ N2 ) ) ) ) ) ).
% one_less_power_int
thf(fact_4305_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N2 )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% power_int_add
thf(fact_4306_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,G: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A3 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_4307_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A4: A,N2: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A4 ) @ N2 )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_int @ A @ A4 @ N2 ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_4308_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N: int,A4: A] :
( ( ord_less_eq @ int @ N2 @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( A4
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A4 @ N ) @ ( power_int @ A @ A4 @ N2 ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_4309_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_4310_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ M @ N2 ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_4311_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ int @ M @ N2 ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_4312_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N2
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_minus_mult
thf(fact_4313_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_4314_multiset_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R3: A > B > $o] :
( relcompp @ ( multiset @ A ) @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ B )
@ ( conversep @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ A )
@ ( bNF_Grp @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ A )
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X2: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ B )
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X2: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% multiset.rel_compp_Grp
thf(fact_4315_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R4: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z5: D] : Y4 = Z5
@ R4 )
= ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_4316_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K5: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K5 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_4317_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_4318_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R4: A > C > $o,S: C > B > $o,T3: C > B > $o] :
( ( relcompp @ A @ C @ B @ R4 @ ( sup_sup @ ( C > B > $o ) @ S @ T3 ) )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ R4 @ S ) @ ( relcompp @ A @ C @ B @ R4 @ T3 ) ) ) ).
% relcompp_distrib
thf(fact_4319_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: A > C > $o,T3: A > C > $o,R4: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( sup_sup @ ( A > C > $o ) @ S @ T3 ) @ R4 )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ S @ R4 ) @ ( relcompp @ A @ C @ B @ T3 @ R4 ) ) ) ).
% relcompp_distrib2
thf(fact_4320_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R4: A > C > $o] :
( ( relcompp @ A @ C @ B @ R4 @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_4321_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R4: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R4 )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_4322_converse__relcompp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: B > C > $o,S2: C > A > $o] :
( ( conversep @ B @ A @ ( relcompp @ B @ C @ A @ R2 @ S2 ) )
= ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ S2 ) @ ( conversep @ B @ C @ R2 ) ) ) ).
% converse_relcompp
thf(fact_4323_funpow__mult,axiom,
! [A: $tType,N2: nat,M: nat,F2: A > A] :
( ( compow @ ( A > A ) @ N2 @ ( compow @ ( A > A ) @ M @ F2 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N2 ) @ F2 ) ) ).
% funpow_mult
thf(fact_4324_group_Oleft__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A,C2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A4 @ B3 )
= ( F2 @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% group.left_cancel
thf(fact_4325_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_4326_group_Oright__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,B3: A,A4: A,C2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ B3 @ A4 )
= ( F2 @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ).
% group.right_cancel
thf(fact_4327_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_4328_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_4329_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_4330_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_4331_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_4332_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_4333_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S2: B > C > $o,A1: A,A22: C] :
( ( relcompp @ A @ B @ C @ R2 @ S2 @ A1 @ A22 )
=> ~ ! [B2: B] :
( ( R2 @ A1 @ B2 )
=> ~ ( S2 @ B2 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_4334_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R5: A > B > $o,S7: B > C > $o,A15: A,A24: C] :
? [A8: A,B6: B,C5: C] :
( ( A15 = A8 )
& ( A24 = C5 )
& ( R5 @ A8 @ B6 )
& ( S7 @ B6 @ C5 ) ) ) ) ).
% relcompp.simps
thf(fact_4335_OO__eq,axiom,
! [B: $tType,A: $tType,R4: A > B > $o] :
( ( relcompp @ A @ B @ B @ R4
@ ^ [Y4: B,Z5: B] : Y4 = Z5 )
= R4 ) ).
% OO_eq
thf(fact_4336_eq__OO,axiom,
! [B: $tType,A: $tType,R4: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ R4 )
= R4 ) ).
% eq_OO
thf(fact_4337_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S2: B > C > $o,A4: A,C2: C] :
( ( relcompp @ A @ B @ C @ R2 @ S2 @ A4 @ C2 )
=> ~ ! [B2: B] :
( ( R2 @ A4 @ B2 )
=> ~ ( S2 @ B2 @ C2 ) ) ) ).
% relcomppE
thf(fact_4338_relcomppI,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,A4: A,B3: B,S2: B > C > $o,C2: C] :
( ( R2 @ A4 @ B3 )
=> ( ( S2 @ B3 @ C2 )
=> ( relcompp @ A @ B @ C @ R2 @ S2 @ A4 @ C2 ) ) ) ).
% relcomppI
thf(fact_4339_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R3: A > B > $o,S6: B > C > $o,A8: A,C5: C] :
? [B6: B] :
( ( R3 @ A8 @ B6 )
& ( S6 @ B6 @ C5 ) ) ) ) ).
% relcompp_apply
thf(fact_4340_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R2: A > D > $o,S2: D > C > $o,T5: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R2 @ S2 ) @ T5 )
= ( relcompp @ A @ D @ B @ R2 @ ( relcompp @ D @ C @ B @ S2 @ T5 ) ) ) ).
% relcompp_assoc
thf(fact_4341_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: B > A,R2: C > A > $o,S2: A > D > $o,A4: C,C2: D] :
( ! [Y2: A] :
? [X6: B] :
( Y2
= ( F2 @ X6 ) )
=> ( ( relcompp @ C @ A @ D @ R2 @ S2 @ A4 @ C2 )
=> ~ ! [B2: B] :
( ( R2 @ A4 @ ( F2 @ B2 ) )
=> ~ ( S2 @ ( F2 @ B2 ) @ C2 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_4342_rel__filter__distr,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A > C > $o,B5: C > B > $o] :
( ( relcompp @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( rel_filter @ A @ C @ A3 ) @ ( rel_filter @ C @ B @ B5 ) )
= ( rel_filter @ A @ B @ ( relcompp @ A @ C @ B @ A3 @ B5 ) ) ) ).
% rel_filter_distr
thf(fact_4343_leq__OOI,axiom,
! [A: $tType,R4: A > A > $o] :
( ( R4
= ( ^ [Y4: A,Z5: A] : Y4 = Z5 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R4 @ ( relcompp @ A @ A @ A @ R4 @ R4 ) ) ) ).
% leq_OOI
thf(fact_4344_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R7: A > B > $o,R2: A > B > $o,S3: B > C > $o,S2: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R7 @ R2 )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S3 @ S2 )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R7 @ S3 ) @ ( relcompp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% relcompp_mono
thf(fact_4345_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q2: B > C > $o,A4: A,C2: C] :
( ( relcompp @ A @ B @ C @ P @ Q2 @ A4 @ C2 )
=> ( ( P @ A4 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q2 @ A4 @ C2 ) )
& ( Q2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q2 @ A4 @ C2 ) @ C2 ) ) ) ).
% pick_middlep
thf(fact_4346_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_4347_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 )
@ ^ [X2: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X2 @ Y3 ) @ S2 ) )
= ( ^ [X2: A,Y3: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y3 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_4348_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q2 ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P ) ) ) ) ).
% fstOp_in
thf(fact_4349_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q2 ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_4350_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R2: D > A > C > $o,I: set @ D,S2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R2 @ I ) ) @ S2 )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I4: D] : ( relcompp @ A @ C @ B @ ( R2 @ I4 ) @ S2 )
@ I ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_4351_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S2: A > C > $o,R2: D > C > B > $o,I: set @ D] :
( ( relcompp @ A @ C @ B @ S2 @ ( complete_Sup_Sup @ ( C > B > $o ) @ ( image2 @ D @ ( C > B > $o ) @ R2 @ I ) ) )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I4: D] : ( relcompp @ A @ C @ B @ S2 @ ( R2 @ I4 ) )
@ I ) ) ) ).
% relcompp_SUP_distrib
thf(fact_4352_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P3: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P3 ) @ P3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P3 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4353_relpowp__bot,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A > $o ) @ N2 @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_4354_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S7: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 )
@ ^ [X2: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X2 @ Y3 ) @ S7 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_4355_OO__Grp__alt,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ C,F2: C > A,G: C > B] :
( ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ ( bNF_Grp @ C @ A @ A3 @ F2 ) ) @ ( bNF_Grp @ C @ B @ A3 @ G ) )
= ( ^ [X2: A,Y3: B] :
? [Z3: C] :
( ( member @ C @ Z3 @ A3 )
& ( ( F2 @ Z3 )
= X2 )
& ( ( G @ Z3 )
= Y3 ) ) ) ) ).
% OO_Grp_alt
thf(fact_4356_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] : ( compow @ ( A > A ) @ N4 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4357_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_4358_Grp__UNIV__id,axiom,
! [A: $tType,F2: A > A] :
( ( F2
= ( id @ A ) )
=> ( ( relcompp @ A @ A @ A @ ( conversep @ A @ A @ ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) )
= ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ).
% Grp_UNIV_id
thf(fact_4359_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q2: A > A] :
( ( order_mono @ A @ A @ Q2 )
=> ( order_mono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q2 @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4360_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N2: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N2 @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_4361_vimage2p__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ C @ B @ D @ $o )
= ( ^ [F4: A > C,G4: B > D,P2: C > D > $o] : ( relcompp @ A @ C @ B @ ( bNF_Grp @ A @ C @ ( top_top @ ( set @ A ) ) @ F4 ) @ ( relcompp @ C @ D @ B @ P2 @ ( conversep @ B @ D @ ( bNF_Grp @ B @ D @ ( top_top @ ( set @ B ) ) @ G4 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_4362_vimage2p__relcompp__converse,axiom,
! [E: $tType,C: $tType,D: $tType,A: $tType,F3: $tType,B: $tType,Rep2: A > B,Abs2: B > A,F2: C > E,G: D > F3,R4: B > E > $o,S: B > F3 > $o] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ E @ D @ F3 @ $o @ F2 @ G @ ( relcompp @ E @ B @ F3 @ ( conversep @ B @ E @ R4 ) @ S ) )
= ( relcompp @ C @ A @ D @ ( conversep @ A @ C @ ( bNF_vimage2p @ A @ B @ C @ E @ $o @ Rep2 @ F2 @ R4 ) ) @ ( bNF_vimage2p @ A @ B @ D @ F3 @ $o @ Rep2 @ G @ S ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_4363_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_4364_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P: A > C > $o,Q2: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F2 @ ( relcompp @ A @ C @ B @ P @ Q2 ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q2 ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_4365_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_4366_less__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A3 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ( ord_less @ ( filter @ A ) )
@ ( ord_less @ ( filter @ B ) ) ) ) ).
% less_filter_parametric
thf(fact_4367_Powp__Pow__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( powp @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( ^ [X2: set @ A] : ( member @ ( set @ A ) @ X2 @ ( pow2 @ A @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_4368_typedef__bi__unique,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B,A3: set @ A,T3: A > B > $o] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ A3 )
=> ( ( T3
= ( ^ [X2: A,Y3: B] :
( X2
= ( Rep2 @ Y3 ) ) ) )
=> ( bi_unique @ A @ B @ T3 ) ) ) ).
% typedef_bi_unique
thf(fact_4369_bi__unique__rel__filter,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bi_unique @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) ) ).
% bi_unique_rel_filter
thf(fact_4370_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_4371_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_4372_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A5: set @ A,F12: B > C,F26: D > C,P13: A > B,P23: A > D] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( F12 @ ( P13 @ X2 ) )
= ( F26 @ ( P23 @ X2 ) ) ) ) ) ) ).
% csquare_def
thf(fact_4373_Grp__fst__snd,axiom,
! [B: $tType,A: $tType,R4: A > B > $o] :
( ( relcompp @ A @ ( product_prod @ A @ B ) @ B @ ( conversep @ ( product_prod @ A @ B ) @ A @ ( bNF_Grp @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_fst @ A @ B ) ) ) @ ( bNF_Grp @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_snd @ A @ B ) ) )
= R4 ) ).
% Grp_fst_snd
thf(fact_4374_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_4375_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_4376_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_4377_Powp__mono,axiom,
! [A: $tType,A3: A > $o,B5: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A3 @ B5 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A3 ) @ ( powp @ A @ B5 ) ) ) ).
% Powp_mono
thf(fact_4378_le__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A3 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ( ord_less_eq @ ( filter @ A ) )
@ ( ord_less_eq @ ( filter @ B ) ) ) ) ).
% le_filter_parametric
thf(fact_4379_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,B2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B2 ) @ ( 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 @ B2 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% lfp_induct2
thf(fact_4380_Powp__def,axiom,
! [A: $tType] :
( ( powp @ A )
= ( ^ [A5: A > $o,B8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ B8 )
=> ( A5 @ X2 ) ) ) ) ).
% Powp_def
thf(fact_4381_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( inf_inf @ ( filter @ A ) ) @ ( inf_inf @ ( filter @ B ) ) ) ) ) ).
% inf_filter_parametric
thf(fact_4382_relpow__fun__conv,axiom,
! [A: $tType,A4: A,B3: A,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A4 )
& ( ( F4 @ N2 )
= B3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_4383_prod_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R13: A > C > $o,R24: B > D > $o] :
( relcompp @ ( product_prod @ A @ B ) @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( conversep @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R13 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R24 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R13 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R24 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% prod.rel_compp_Grp
thf(fact_4384_prod_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R13: A > C > $o,R24: B > D > $o,A8: product_prod @ A @ B,B6: product_prod @ C @ D] :
? [Z3: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z3
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R13 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R24 ) ) ) ) ) )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z3 )
= A8 )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z3 )
= B6 ) ) ) ) ).
% prod.in_rel
thf(fact_4385_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A4: A,B3: C,C2: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A4 @ B3 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
= ( ( R12 @ A4 @ C2 )
& ( R23 @ B3 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_4386_relpow__Suc__D2_H,axiom,
! [A: $tType,N2: nat,R4: set @ ( product_prod @ A @ A ),X6: A,Y6: A,Z7: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z7 ) @ R4 ) )
=> ? [W2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ W2 ) @ R4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W2 @ Z7 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_4387_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,B2: B,C3: C] :
( ( A1
= ( product_Pair @ A @ C @ A6 @ C3 ) )
=> ! [D2: D] :
( ( A22
= ( product_Pair @ B @ D @ B2 @ D2 ) )
=> ( ( R12 @ A6 @ B2 )
=> ~ ( R23 @ C3 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_4388_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,A15: product_prod @ A @ C,A24: product_prod @ B @ D] :
? [A8: A,B6: B,C5: C,D5: D] :
( ( A15
= ( product_Pair @ A @ C @ A8 @ C5 ) )
& ( A24
= ( product_Pair @ B @ D @ B6 @ D5 ) )
& ( R13 @ A8 @ B6 )
& ( R24 @ C5 @ D5 ) ) ) ) ).
% rel_prod.simps
thf(fact_4389_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A4: A,B3: B,R23: C > D > $o,C2: C,D3: D] :
( ( R12 @ A4 @ B3 )
=> ( ( R23 @ C2 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A4 @ C2 ) @ ( product_Pair @ B @ D @ B3 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_4390_bi__total__rel__filter,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bi_total @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) ) ).
% bi_total_rel_filter
thf(fact_4391_rel__prod__sel,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R13: A > B > $o,R24: C > D > $o,P5: product_prod @ A @ C,Q6: product_prod @ B @ D] :
( ( R13 @ ( product_fst @ A @ C @ P5 ) @ ( product_fst @ B @ D @ Q6 ) )
& ( R24 @ ( product_snd @ A @ C @ P5 ) @ ( product_snd @ B @ D @ Q6 ) ) ) ) ) ).
% rel_prod_sel
thf(fact_4392_snd__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: A > C > $o,B5: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D @ ( basic_rel_prod @ A @ C @ B @ D @ A3 @ B5 ) @ B5 @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) ).
% snd_transfer
thf(fact_4393_relpow__0__I,axiom,
! [A: $tType,X: A,R4: 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 ) @ R4 ) ) ).
% relpow_0_I
thf(fact_4394_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R4: 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 ) @ R4 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_4395_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A ),Z2: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R4 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_4396_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R4 ) )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_4397_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R4 ) )
=> ? [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_4398_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N2: nat,R4: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R4 ) ) ) ) ).
% relpow_Suc_I
thf(fact_4399_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R4 ) )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R4 ) ) ) ).
% relpow_Suc_E
thf(fact_4400_prod__filter__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R4: A > B > $o,S: C > D > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ C ) > ( filter @ ( product_prod @ A @ C ) ) ) @ ( ( filter @ D ) > ( filter @ ( product_prod @ B @ D ) ) ) @ ( rel_filter @ A @ B @ R4 ) @ ( bNF_rel_fun @ ( filter @ C ) @ ( filter @ D ) @ ( filter @ ( product_prod @ A @ C ) ) @ ( filter @ ( product_prod @ B @ D ) ) @ ( rel_filter @ C @ D @ S ) @ ( rel_filter @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( basic_rel_prod @ A @ B @ C @ D @ R4 @ S ) ) ) @ ( prod_filter @ A @ C ) @ ( prod_filter @ B @ D ) ) ).
% prod_filter_parametric
thf(fact_4401_top__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( rel_filter @ A @ B @ A3 @ ( top_top @ ( filter @ A ) ) @ ( top_top @ ( filter @ B ) ) ) ) ).
% top_filter_parametric
thf(fact_4402_relpowp__relpow__eq,axiom,
! [A: $tType,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N2
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R4 ) )
= ( ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_4403_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y2: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R4 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_4404_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y2: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R4 ) ) ) ) ) ).
% relpow_E
thf(fact_4405_relpow__empty,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_4406_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B5: 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 ) @ B5 @ ( basic_rel_prod @ A @ B @ C @ D @ A3 @ B5 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_4407_curry__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F3: $tType,E: $tType,D: $tType,A3: A > D > $o,B5: B > E > $o,C6: C > F3 > $o] : ( bNF_rel_fun @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F3 ) @ ( A > B > C ) @ ( D > E > F3 ) @ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F3 @ ( basic_rel_prod @ A @ D @ B @ E @ A3 @ B5 ) @ C6 ) @ ( bNF_rel_fun @ A @ D @ ( B > C ) @ ( E > F3 ) @ A3 @ ( bNF_rel_fun @ B @ E @ C @ F3 @ B5 @ C6 ) ) @ ( product_curry @ A @ B @ C ) @ ( product_curry @ D @ E @ F3 ) ) ).
% curry_transfer
thf(fact_4408_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X: set @ ( A > nat )] :
( ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) ) )
@ X
@ X )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ ( A > nat ) @ ( multiset @ A ) @ ( abs_multiset @ A ) @ X ) )
= ( abs_multiset @ A
@ ^ [I4: A] :
( if @ nat
@ ( X
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ X ) ) ) ) ) ) ).
% Inf_multiset.abs_eq
thf(fact_4409_irreflp__irrefl__eq,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A8: A,B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R4 ) )
= ( irrefl @ A @ R4 ) ) ).
% irreflp_irrefl_eq
thf(fact_4410_equivp__equiv,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A3 )
= ( equiv_equivp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ A3 ) ) ) ).
% equivp_equiv
thf(fact_4411_is__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( ( A > $o ) > $o ) @ ( ( B > $o ) > $o ) @ $o @ $o
@ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5 )
@ ^ [Y4: $o,Z5: $o] : Y4 = Z5
@ ( is_filter @ A )
@ ( is_filter @ B ) ) ) ) ).
% is_filter_parametric
thf(fact_4412_eq__onp__top__eq__eq,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
= ( ^ [Y4: A,Z5: A] : Y4 = Z5 ) ) ).
% eq_onp_top_eq_eq
thf(fact_4413_eq__onp__same__args,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( bNF_eq_onp @ A @ P @ X @ X )
= ( P @ X ) ) ).
% eq_onp_same_args
thf(fact_4414_eq__onp__to__eq,axiom,
! [A: $tType,P: A > $o,X: A,Y: A] :
( ( bNF_eq_onp @ A @ P @ X @ Y )
=> ( X = Y ) ) ).
% eq_onp_to_eq
thf(fact_4415_eq__onp__mono0,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ! [X6: A] :
( ( member @ A @ X6 @ A3 )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( bNF_eq_onp @ A @ P @ X6 @ Xa2 )
=> ( bNF_eq_onp @ A @ Q2 @ X6 @ Xa2 ) ) ) ) ) ).
% eq_onp_mono0
thf(fact_4416_eq__onp__eqD,axiom,
! [A: $tType,P: A > $o,Q2: A > A > $o,X: A] :
( ( ( bNF_eq_onp @ A @ P )
= Q2 )
=> ( ( P @ X )
= ( Q2 @ X @ X ) ) ) ).
% eq_onp_eqD
thf(fact_4417_rel__setI,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B,R4: A > B > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B5 )
& ( R4 @ X3 @ Xa2 ) ) )
=> ( ! [Y2: B] :
( ( member @ B @ Y2 @ B5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A3 )
& ( R4 @ X6 @ Y2 ) ) )
=> ( bNF_rel_set @ A @ B @ R4 @ A3 @ B5 ) ) ) ).
% rel_setI
thf(fact_4418_irreflp__def,axiom,
! [A: $tType] :
( ( irreflp @ A )
= ( ^ [R3: A > A > $o] :
! [A8: A] :
~ ( R3 @ A8 @ A8 ) ) ) ).
% irreflp_def
thf(fact_4419_irreflpI,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [A6: A] :
~ ( R4 @ A6 @ A6 )
=> ( irreflp @ A @ R4 ) ) ).
% irreflpI
thf(fact_4420_is__filter__def,axiom,
! [A: $tType] :
( ( is_filter @ A )
= ( ^ [F7: ( A > $o ) > $o] :
( ( F7
@ ^ [X2: A] : $true )
& ! [P2: A > $o] :
( ( F7 @ P2 )
=> ! [Q: A > $o] :
( ( F7 @ Q )
=> ( F7
@ ^ [X2: A] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) ) )
& ! [P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ( F7 @ P2 )
=> ( F7 @ Q ) ) ) ) ) ) ).
% is_filter_def
thf(fact_4421_is__filter_Ointro,axiom,
! [A: $tType,F5: ( A > $o ) > $o] :
( ( F5
@ ^ [X2: A] : $true )
=> ( ! [P4: A > $o] :
( ( F5 @ P4 )
=> ! [Q3: A > $o] :
( ( F5 @ Q3 )
=> ( F5
@ ^ [X2: A] :
( ( P4 @ X2 )
& ( Q3 @ X2 ) ) ) ) )
=> ( ! [P4: A > $o,Q3: A > $o] :
( ! [X6: A] :
( ( P4 @ X6 )
=> ( Q3 @ X6 ) )
=> ( ( F5 @ P4 )
=> ( F5 @ Q3 ) ) )
=> ( is_filter @ A @ F5 ) ) ) ) ).
% is_filter.intro
thf(fact_4422_is__filter_Omono,axiom,
! [A: $tType,F5: ( A > $o ) > $o,P: A > $o,Q2: A > $o] :
( ( is_filter @ A @ F5 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ( F5 @ P )
=> ( F5 @ Q2 ) ) ) ) ).
% is_filter.mono
thf(fact_4423_is__filter_Oconj,axiom,
! [A: $tType,F5: ( A > $o ) > $o,P: A > $o,Q2: A > $o] :
( ( is_filter @ A @ F5 )
=> ( ( F5 @ P )
=> ( ( F5 @ Q2 )
=> ( F5
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% is_filter.conj
thf(fact_4424_is__filter_OTrue,axiom,
! [A: $tType,F5: ( A > $o ) > $o] :
( ( is_filter @ A @ F5 )
=> ( F5
@ ^ [X2: A] : $true ) ) ).
% is_filter.True
thf(fact_4425_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu: A] : $true )
= ( ^ [Y4: A,Z5: A] : Y4 = Z5 ) ) ).
% eq_onp_True
thf(fact_4426_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R3: A > $o,X2: A,Y3: A] :
( ( R3 @ X2 )
& ( X2 = Y3 ) ) ) ) ).
% eq_onp_def
thf(fact_4427_irreflp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% irreflp_less
thf(fact_4428_principal__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ A3 ) @ ( principal @ A ) @ ( principal @ B ) ) ).
% principal_parametric
thf(fact_4429_rel__set__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_set @ A @ B )
= ( ^ [R3: A > B > $o,A5: set @ A,B8: set @ B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ? [Y3: B] :
( ( member @ B @ Y3 @ B8 )
& ( R3 @ X2 @ Y3 ) ) )
& ! [X2: B] :
( ( member @ B @ X2 @ B8 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A5 )
& ( R3 @ Y3 @ X2 ) ) ) ) ) ) ).
% rel_set_def
thf(fact_4430_eq__onp__Grp,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [P2: A > $o] : ( bNF_Grp @ A @ A @ ( collect @ A @ P2 ) @ ( id @ A ) ) ) ) ).
% eq_onp_Grp
thf(fact_4431_Abs__filter__inject,axiom,
! [A: $tType,X: ( A > $o ) > $o,Y: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ X @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ( ( abs_filter @ A @ X )
= ( abs_filter @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_filter_inject
thf(fact_4432_Abs__filter__induct,axiom,
! [A: $tType,P: ( filter @ A ) > $o,X: filter @ A] :
( ! [Y2: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y2 @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( P @ ( abs_filter @ A @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_filter_induct
thf(fact_4433_Abs__filter__cases,axiom,
! [A: $tType,X: filter @ A] :
~ ! [Y2: ( A > $o ) > $o] :
( ( X
= ( abs_filter @ A @ Y2 ) )
=> ~ ( member @ ( ( A > $o ) > $o ) @ Y2 @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) ) ) ).
% Abs_filter_cases
thf(fact_4434_eq__onp__mono__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P ) @ ( bNF_eq_onp @ A @ Q2 ) )
= ( ord_less_eq @ ( A > $o ) @ P @ Q2 ) ) ).
% eq_onp_mono_iff
thf(fact_4435_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% irreflp_greater
thf(fact_4436_is__filter__map__filter__on,axiom,
! [A: $tType,B: $tType,F2: B > A,X4: set @ B,F5: filter @ B] :
( ( is_filter @ A
@ ^ [P2: A > $o] :
( eventually @ B
@ ^ [X2: B] :
( ( P2 @ ( F2 @ X2 ) )
& ( member @ B @ X2 @ X4 ) )
@ F5 ) )
= ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ X4 )
@ F5 ) ) ).
% is_filter_map_filter_on
thf(fact_4437_UNIV__typedef__to__equivp,axiom,
! [B: $tType,A: $tType,Rep2: B > A,Abs2: A > B] :
( ( type_definition @ B @ A @ Rep2 @ Abs2 @ ( top_top @ ( set @ A ) ) )
=> ( equiv_equivp @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) ) ).
% UNIV_typedef_to_equivp
thf(fact_4438_eventually__Abs__filter,axiom,
! [A: $tType,F5: ( A > $o ) > $o,P: A > $o] :
( ( is_filter @ A @ F5 )
=> ( ( eventually @ A @ P @ ( abs_filter @ A @ F5 ) )
= ( F5 @ P ) ) ) ).
% eventually_Abs_filter
thf(fact_4439_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 )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) ) )
@ ^ [A5: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A5
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A5 ) ) )
@ ^ [A5: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A5
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A5 ) ) ) ) ).
% Inf_multiset.rsp
thf(fact_4440_repeat__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) ) ) )
@ ^ [N4: nat,M5: A > nat,A8: A] : ( times_times @ nat @ N4 @ ( M5 @ A8 ) )
@ ^ [N4: nat,M5: A > nat,A8: A] : ( times_times @ nat @ N4 @ ( M5 @ A8 ) ) ) ).
% repeat_mset.rsp
thf(fact_4441_repeat__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X2 ) ) ) )
@ X
@ X )
=> ( ( repeat_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A8: A] : ( times_times @ nat @ Xa @ ( X @ A8 ) ) ) ) ) ).
% repeat_mset.abs_eq
thf(fact_4442_Lifting__Set_Oinsert__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ A3 @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( bNF_rel_set @ A @ B @ A3 ) ) @ ( insert2 @ A ) @ ( insert2 @ B ) ) ).
% Lifting_Set.insert_transfer
thf(fact_4443_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_4444_Abs__filter__inverse,axiom,
! [A: $tType,Y: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ( rep_filter @ A @ ( abs_filter @ A @ Y ) )
= Y ) ) ).
% Abs_filter_inverse
thf(fact_4445_repeat__mset__right,axiom,
! [A: $tType,A4: nat,B3: nat,A3: multiset @ A] :
( ( repeat_mset @ A @ A4 @ ( repeat_mset @ A @ B3 @ A3 ) )
= ( repeat_mset @ A @ ( times_times @ nat @ A4 @ B3 ) @ A3 ) ) ).
% repeat_mset_right
thf(fact_4446_Rep__filter__iff__eventually,axiom,
! [A: $tType] :
( ( rep_filter @ A )
= ( ^ [F7: filter @ A,P2: A > $o] : ( eventually @ A @ P2 @ F7 ) ) ) ).
% Rep_filter_iff_eventually
thf(fact_4447_count__repeat__mset,axiom,
! [A: $tType,I2: nat,A3: multiset @ A,A4: A] :
( ( count @ A @ ( repeat_mset @ A @ I2 @ A3 ) @ A4 )
= ( times_times @ nat @ I2 @ ( count @ A @ A3 @ A4 ) ) ) ).
% count_repeat_mset
thf(fact_4448_Rep__filter__inject,axiom,
! [A: $tType,X: filter @ A,Y: filter @ A] :
( ( ( rep_filter @ A @ X )
= ( rep_filter @ A @ Y ) )
= ( X = Y ) ) ).
% Rep_filter_inject
thf(fact_4449_Sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( set @ ( filter @ A ) ) @ ( set @ ( filter @ B ) ) @ ( filter @ A ) @ ( filter @ B ) @ ( bNF_rel_set @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( complete_Sup_Sup @ ( filter @ A ) ) @ ( complete_Sup_Sup @ ( filter @ B ) ) ) ).
% Sup_filter_parametric
thf(fact_4450_is__filter__Rep__filter,axiom,
! [A: $tType,F5: filter @ A] : ( is_filter @ A @ ( rep_filter @ A @ F5 ) ) ).
% is_filter_Rep_filter
thf(fact_4451_Rep__filter__inverse,axiom,
! [A: $tType,X: filter @ A] :
( ( abs_filter @ A @ ( rep_filter @ A @ X ) )
= X ) ).
% Rep_filter_inverse
thf(fact_4452_repeat__mset_Orep__eq,axiom,
! [A: $tType,X: nat,Xa: multiset @ A] :
( ( count @ A @ ( repeat_mset @ A @ X @ Xa ) )
= ( ^ [A8: A] : ( times_times @ nat @ X @ ( count @ A @ Xa @ A8 ) ) ) ) ).
% repeat_mset.rep_eq
thf(fact_4453_Rep__filter,axiom,
! [A: $tType,X: filter @ A] : ( member @ ( ( A > $o ) > $o ) @ ( rep_filter @ A @ X ) @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) ) ).
% Rep_filter
thf(fact_4454_Rep__filter__cases,axiom,
! [A: $tType,Y: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ~ ! [X3: filter @ A] :
( Y
!= ( rep_filter @ A @ X3 ) ) ) ).
% Rep_filter_cases
thf(fact_4455_Rep__filter__induct,axiom,
! [A: $tType,Y: ( A > $o ) > $o,P: ( ( A > $o ) > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ! [X3: filter @ A] : ( P @ ( rep_filter @ A @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_filter_induct
thf(fact_4456_Inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( set @ ( filter @ A ) ) @ ( set @ ( filter @ B ) ) @ ( filter @ A ) @ ( filter @ B ) @ ( bNF_rel_set @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( complete_Inf_Inf @ ( filter @ A ) ) @ ( complete_Inf_Inf @ ( filter @ B ) ) ) ) ) ).
% Inf_filter_parametric
thf(fact_4457_Abs__filter__inverse_H,axiom,
! [A: $tType,F5: ( A > $o ) > $o] :
( ( is_filter @ A @ F5 )
=> ( ( rep_filter @ A @ ( abs_filter @ A @ F5 ) )
= F5 ) ) ).
% Abs_filter_inverse'
thf(fact_4458_type__definition__filter,axiom,
! [A: $tType] : ( type_definition @ ( filter @ A ) @ ( ( A > $o ) > $o ) @ ( rep_filter @ A ) @ ( abs_filter @ A ) @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) ) ).
% type_definition_filter
thf(fact_4459_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 )
@ ^ [A5: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A5
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A5 ) ) )
@ ( complete_Inf_Inf @ ( multiset @ A ) ) ) ).
% Inf_multiset.transfer
thf(fact_4460_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% in_chain_finite
thf(fact_4461_cofinite__bot,axiom,
! [A: $tType] :
( ( ( cofinite @ A )
= ( bot_bot @ ( filter @ A ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% cofinite_bot
thf(fact_4462_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_4463_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_4464_cofinite__eq__sequentially,axiom,
( ( cofinite @ nat )
= ( at_top @ nat ) ) ).
% cofinite_eq_sequentially
thf(fact_4465_ccpo__Sup__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B5 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% ccpo_Sup_mono
thf(fact_4466_INFM__nat__inductI,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [I3: nat] :
( ( P @ I3 )
=> ? [J7: nat] :
( ( ord_less @ nat @ I3 @ J7 )
& ( P @ J7 )
& ( Q2 @ J7 ) ) )
=> ( frequently @ nat @ Q2 @ ( cofinite @ nat ) ) ) ) ).
% INFM_nat_inductI
thf(fact_4467_eventually__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) ).
% eventually_cofinite
thf(fact_4468_frequently__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% frequently_cofinite
thf(fact_4469_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_4470_cofinite__def,axiom,
! [A: $tType] :
( ( cofinite @ A )
= ( abs_filter @ A
@ ^ [P2: A > $o] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P2 @ X2 ) ) ) ) ) ).
% cofinite_def
thf(fact_4471_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,N2: nat] :
( ( divide_divide @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ A4 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_4472_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N4 @ A8 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_4473_repeat__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y4: nat,Z5: nat] : Y4 = Z5
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
@ ( pcr_multiset @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) )
@ ^ [N4: nat,M5: A > nat,A8: A] : ( times_times @ nat @ N4 @ ( M5 @ A8 ) )
@ ( repeat_mset @ A ) ) ).
% repeat_mset.transfer
thf(fact_4474_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_4475_wfP__SUP,axiom,
! [B: $tType,A: $tType,R2: A > B > B > $o] :
( ! [I3: A] : ( wfP @ B @ ( R2 @ I3 ) )
=> ( ! [I3: A,J4: A] :
( ( ( R2 @ I3 )
!= ( R2 @ J4 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R2 @ I3 ) ) @ ( rangep @ B @ B @ ( R2 @ J4 ) ) )
= ( bot_bot @ ( B > $o ) ) ) )
=> ( wfP @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% wfP_SUP
thf(fact_4476_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
& ( N2
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ) ).
% bit_double_iff
thf(fact_4477_override__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B,X: A,X4: set @ A] :
( ( override_on @ A @ B @ F2 @ G @ ( insert2 @ A @ X @ X4 ) )
= ( fun_upd @ A @ B @ ( override_on @ A @ B @ F2 @ G @ X4 ) @ X @ ( G @ X ) ) ) ).
% override_on_insert
thf(fact_4478_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ( override_on @ A @ B @ F2 @ G @ ( bot_bot @ ( set @ A ) ) )
= F2 ) ).
% override_on_emptyset
thf(fact_4479_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ).
% bit_minus_1_iff
thf(fact_4480_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: A] :
( ( domainp @ A @ B @ R2 @ A4 )
=> ~ ! [B2: B] :
~ ( R2 @ A4 @ B2 ) ) ).
% Domainp.cases
thf(fact_4481_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R5: A > B > $o,A8: A] :
? [B6: A,C5: B] :
( ( A8 = B6 )
& ( R5 @ B6 @ C5 ) ) ) ) ).
% Domainp.simps
thf(fact_4482_DomainPI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,A4: A,B3: B] :
( ( R2 @ A4 @ B3 )
=> ( domainp @ A @ B @ R2 @ A4 ) ) ).
% DomainPI
thf(fact_4483_DomainpE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: A] :
( ( domainp @ A @ B @ R2 @ A4 )
=> ~ ! [B2: B] :
~ ( R2 @ A4 @ B2 ) ) ).
% DomainpE
thf(fact_4484_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A4 ) @ N2 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% bit_minus_iff
thf(fact_4485_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( domain @ A @ B @ R2 ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_4486_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% Domain_def
thf(fact_4487_wfP__wf__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( wf @ A @ R2 ) ) ).
% wfP_wf_eq
thf(fact_4488_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N2 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ( ord_less @ nat @ N2 @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_4489_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B,X: A,X4: set @ A] :
( ( override_on @ A @ B @ F2 @ G @ ( insert2 @ A @ X @ X4 ) )
= ( override_on @ A @ B @ ( fun_upd @ A @ B @ F2 @ X @ ( G @ X ) ) @ G @ X4 ) ) ).
% override_on_insert'
thf(fact_4490_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X2: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_unit_def
thf(fact_4491_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F4: A > B,R3: set @ ( product_prod @ A @ A ),X2: A,Y3: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R3 ) @ ( F4 @ Y3 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_4492_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: A] :
( collect @ A
@ ^ [B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R5 ) ) ) ) ).
% above_def
thf(fact_4493_ID_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( bNF_id_bnf @ ( A > B > $o ) )
= ( ^ [R3: A > B > $o,A8: A,B6: 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 @ R3 ) ) ) ) )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > A ) @ ( product_fst @ A @ B ) @ Z3 )
= A8 )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > B ) @ ( product_snd @ A @ B ) @ Z3 )
= B6 ) ) ) ) ).
% ID.in_rel
thf(fact_4494_cuts__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,R4: set @ ( product_prod @ A @ A ),X: A,G: A > B] :
( ( ( cut @ A @ B @ F2 @ R4 @ X )
= ( cut @ A @ B @ G @ R4 @ X ) )
= ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R4 )
=> ( ( F2 @ Y3 )
= ( G @ Y3 ) ) ) ) ) ).
% cuts_eq
thf(fact_4495_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A4: A,R4: set @ ( product_prod @ A @ A ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A4 ) @ R4 )
=> ( ( cut @ A @ B @ F2 @ R4 @ A4 @ X )
= ( F2 @ X ) ) ) ).
% cut_apply
thf(fact_4496_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya2: A,Y: B,Xa: B,R4: 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 ) ) ) )
=> ( ( R4 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_id_bnf @ ( A > B > $o ) @ R4 @ X @ Y )
= ( bNF_id_bnf @ ( A > B > $o ) @ Ra2 @ Ya2 @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_4497_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,X: A,Y: B,Ra2: A > B > $o] :
( ( bNF_id_bnf @ ( A > B > $o ) @ R4 @ 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 ) ) ) )
=> ( ( R4 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_id_bnf @ ( A > B > $o ) @ Ra2 @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_4498_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_4499_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
! [A: $tType] : ( type_definition @ A @ A @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ A ) @ ( top_top @ ( set @ A ) ) ) ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_4500_ctor__rec__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R4: A > C > $o,S: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( A > B ) @ ( C > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( bNF_vimage2p @ A @ A @ C @ C @ $o @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ C ) @ R4 ) @ S ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R4 @ S ) @ ( basic_BNF_ctor_rec @ ( A > B ) ) @ ( basic_BNF_ctor_rec @ ( C > D ) ) ) ).
% ctor_rec_transfer
thf(fact_4501_Basic__BNF__LFPs_Oxtor__rel__induct,axiom,
! [B: $tType,A: $tType,R4: A > B > $o,IR: A > B > $o] :
( ! [X3: A,Y2: B] :
( ( bNF_vimage2p @ A @ A @ B @ B @ $o @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ B ) @ R4 @ X3 @ Y2 )
=> ( IR @ ( basic_BNF_xtor @ A @ X3 ) @ ( basic_BNF_xtor @ B @ Y2 ) ) )
=> ( ord_less_eq @ ( A > B > $o ) @ R4 @ IR ) ) ).
% Basic_BNF_LFPs.xtor_rel_induct
thf(fact_4502_Basic__BNF__LFPs_Octor__rec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C] :
( ( comp @ C @ B @ A @ ( basic_BNF_ctor_rec @ ( C > B ) @ F2 ) @ G )
= ( basic_BNF_ctor_rec @ ( A > B ) @ ( comp @ C @ B @ A @ F2 @ ( comp @ A @ C @ A @ ( comp @ C @ C @ A @ ( bNF_id_bnf @ C ) @ G ) @ ( bNF_id_bnf @ A ) ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec_o_map
thf(fact_4503_Basic__BNF__LFPs_Octor__rec__def__alt,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( F2
= ( basic_BNF_ctor_rec @ ( A > B ) @ ( comp @ A @ B @ A @ F2 @ ( bNF_id_bnf @ A ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec_def_alt
thf(fact_4504_Basic__BNF__LFPs_OPair__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair @ A @ B )
= ( ^ [A8: A,B6: B] : ( basic_BNF_xtor @ ( product_prod @ A @ B ) @ ( bNF_id_bnf @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B6 ) ) ) ) ) ).
% Basic_BNF_LFPs.Pair_def_alt
thf(fact_4505_Basic__BNF__LFPs_Oxtor__def,axiom,
! [A: $tType] :
( ( basic_BNF_xtor @ A )
= ( ^ [X2: A] : X2 ) ) ).
% Basic_BNF_LFPs.xtor_def
thf(fact_4506_Basic__BNF__LFPs_Oxtor__map,axiom,
! [A: $tType,B: $tType,F2: B > A,X: B] :
( ( F2 @ ( basic_BNF_xtor @ B @ X ) )
= ( basic_BNF_xtor @ A @ ( F2 @ X ) ) ) ).
% Basic_BNF_LFPs.xtor_map
thf(fact_4507_Basic__BNF__LFPs_Oxtor__rel,axiom,
! [B: $tType,A: $tType,C: $tType,R4: B > C > A,X: B,Y: C] :
( ( R4 @ ( basic_BNF_xtor @ B @ X ) @ ( basic_BNF_xtor @ C @ Y ) )
= ( R4 @ X @ Y ) ) ).
% Basic_BNF_LFPs.xtor_rel
thf(fact_4508_Basic__BNF__LFPs_Oxtor__set,axiom,
! [A: $tType,B: $tType,F2: B > A,X: B] :
( ( F2 @ ( basic_BNF_xtor @ B @ X ) )
= ( F2 @ X ) ) ).
% Basic_BNF_LFPs.xtor_set
thf(fact_4509_Basic__BNF__LFPs_Oxtor__xtor,axiom,
! [A: $tType,X: A] :
( ( basic_BNF_xtor @ A @ ( basic_BNF_xtor @ A @ X ) )
= X ) ).
% Basic_BNF_LFPs.xtor_xtor
thf(fact_4510_Basic__BNF__LFPs_Oxtor__induct,axiom,
! [A: $tType,P: A > $o,Z2: A] :
( ! [X3: A] : ( P @ ( basic_BNF_xtor @ A @ X3 ) )
=> ( P @ Z2 ) ) ).
% Basic_BNF_LFPs.xtor_induct
thf(fact_4511_Basic__BNF__LFPs_Oxtor__inject,axiom,
! [A: $tType,X: A,Y: A] :
( ( ( basic_BNF_xtor @ A @ X )
= ( basic_BNF_xtor @ A @ Y ) )
= ( X = Y ) ) ).
% Basic_BNF_LFPs.xtor_inject
thf(fact_4512_xtor__iff__xtor,axiom,
! [A: $tType,U: A,W: A] :
( ( U
= ( basic_BNF_xtor @ A @ W ) )
= ( ( basic_BNF_xtor @ A @ U )
= W ) ) ).
% xtor_iff_xtor
thf(fact_4513_xtor__map__unique,axiom,
! [B: $tType,A: $tType,U: A > B,F2: A > B] :
( ( ( comp @ A @ B @ A @ U @ ( basic_BNF_xtor @ A ) )
= ( comp @ B @ B @ A @ ( basic_BNF_xtor @ B ) @ F2 ) )
=> ( U = F2 ) ) ).
% xtor_map_unique
thf(fact_4514_Basic__BNF__LFPs_Octor__rec__def,axiom,
! [A: $tType] :
( ( basic_BNF_ctor_rec @ A )
= ( ^ [X2: A] : X2 ) ) ).
% Basic_BNF_LFPs.ctor_rec_def
thf(fact_4515_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_4516_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_4517_ctor__rec__unique,axiom,
! [B: $tType,A: $tType,G: A > A,F2: A > B,S2: A > B] :
( ( G
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ A @ F2 @ ( basic_BNF_xtor @ A ) )
= ( comp @ A @ B @ A @ S2 @ ( comp @ A @ A @ A @ ( comp @ A @ A @ A @ ( bNF_id_bnf @ A ) @ G ) @ ( bNF_id_bnf @ A ) ) ) )
=> ( F2
= ( basic_BNF_ctor_rec @ ( A > B ) @ S2 ) ) ) ) ).
% ctor_rec_unique
thf(fact_4518_Basic__BNF__LFPs_Octor__rec,axiom,
! [B: $tType,A: $tType,G: A > A,F2: A > B,X: A] :
( ( G
= ( id @ A ) )
=> ( ( basic_BNF_ctor_rec @ ( A > B ) @ F2 @ ( basic_BNF_xtor @ A @ X ) )
= ( F2 @ ( comp @ A @ A @ A @ ( comp @ A @ A @ A @ ( bNF_id_bnf @ A ) @ G ) @ ( bNF_id_bnf @ A ) @ X ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec
thf(fact_4519_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P2: A > $o,X2: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P2 @ Y3 ) ) ) ) ).
% ID.pred_set
thf(fact_4520_lcm__altdef__int,axiom,
( ( gcd_lcm @ int )
= ( ^ [A8: int,B6: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A8 ) @ ( abs_abs @ int @ B6 ) ) @ ( gcd_gcd @ int @ A8 @ B6 ) ) ) ) ).
% lcm_altdef_int
thf(fact_4521_times__num__def,axiom,
( ( times_times @ num )
= ( ^ [M4: num,N4: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M4 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).
% times_num_def
thf(fact_4522_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,Y2: A] :
( ( A4
= ( plus_plus @ A @ X3 @ Y2 ) )
=> ( ( neg_numeral_is_num @ A @ X3 )
=> ~ ( neg_numeral_is_num @ A @ Y2 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_4523_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A8: A] :
( ( A8
= ( one_one @ A ) )
| ? [X2: A] :
( ( A8
= ( uminus_uminus @ A @ X2 ) )
& ( neg_numeral_is_num @ A @ X2 ) )
| ? [X2: A,Y3: A] :
( ( A8
= ( plus_plus @ A @ X2 @ Y3 ) )
& ( neg_numeral_is_num @ A @ X2 )
& ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_4524_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_4525_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_4526_nat__of__num__mult,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( times_times @ num @ X @ Y ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_mult
thf(fact_4527_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 )
= ( gcd_lcm @ A @ B3 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_4528_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B3 @ ( times_times @ A @ C2 @ A4 ) )
= ( gcd_lcm @ A @ B3 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_4529_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B3 @ ( divide_divide @ A @ C2 @ A4 ) )
= ( gcd_lcm @ A @ B3 @ C2 ) ) ) ) ).
% lcm_div_unit2
thf(fact_4530_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B3 @ A4 ) @ C2 )
= ( gcd_lcm @ A @ B3 @ C2 ) ) ) ) ).
% lcm_div_unit1
thf(fact_4531_prod__gcd__lcm__int,axiom,
! [M: int,N2: int] :
( ( times_times @ int @ ( abs_abs @ int @ M ) @ ( abs_abs @ int @ N2 ) )
= ( times_times @ int @ ( gcd_gcd @ int @ M @ N2 ) @ ( gcd_lcm @ int @ M @ N2 ) ) ) ).
% prod_gcd_lcm_int
thf(fact_4532_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A5: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A5 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_4533_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_4534_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_4535_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_4536_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_4537_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_4538_Lcm__fin_Oinsert,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 @ A3 ) ) ) ) ).
% Lcm_fin.insert
thf(fact_4539_prod__gcd__lcm__nat,axiom,
( ( times_times @ nat )
= ( ^ [M4: nat,N4: nat] : ( times_times @ nat @ ( gcd_gcd @ nat @ M4 @ N4 ) @ ( gcd_lcm @ nat @ M4 @ N4 ) ) ) ) ).
% prod_gcd_lcm_nat
thf(fact_4540_lcm__code__integer,axiom,
( ( gcd_lcm @ code_integer )
= ( ^ [A8: code_integer,B6: code_integer] : ( divide_divide @ code_integer @ ( times_times @ code_integer @ ( abs_abs @ code_integer @ A8 ) @ ( abs_abs @ code_integer @ B6 ) ) @ ( gcd_gcd @ code_integer @ A8 @ B6 ) ) ) ) ).
% lcm_code_integer
thf(fact_4541_lcm__nat__def,axiom,
( ( gcd_lcm @ nat )
= ( ^ [X2: nat,Y3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X2 @ Y3 ) @ ( gcd_gcd @ nat @ X2 @ Y3 ) ) ) ) ).
% lcm_nat_def
thf(fact_4542_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_finite2 @ A @ A3 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_4543_Lcm__eq__Max__nat,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( M2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ! [M3: nat,N3: nat] :
( ( member @ nat @ M3 @ M2 )
=> ( ( member @ nat @ N3 @ M2 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N3 ) @ M2 ) ) )
=> ( ( gcd_Lcm @ nat @ M2 )
= ( lattic643756798349783984er_Max @ nat @ M2 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_4544_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_4545_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_4546_nat__of__num__sqr,axiom,
! [X: num] :
( ( nat_of_num @ ( sqr @ X ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_sqr
thf(fact_4547_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_4548_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_4549_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_4550_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_4551_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_4552_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_4553_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_4554_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_4555_Lcm__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A4 @ A3 ) )
= ( gcd_lcm @ A @ A4 @ ( gcd_Lcm @ A @ A3 ) ) ) ) ).
% Lcm_insert
thf(fact_4556_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_4557_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_4558_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_4559_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_4560_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_4561_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_4562_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_4563_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A3 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_4564_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_4565_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_4566_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_4567_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_4568_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_4569_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% gcd_mult_right
thf(fact_4570_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_4571_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A4 @ B3 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_4572_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% lcm_mult_right
thf(fact_4573_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_4574_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_4575_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times @ num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_4576_Lcm__in__lcm__closed__set__nat,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( M2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ! [M3: nat,N3: nat] :
( ( member @ nat @ M3 @ M2 )
=> ( ( member @ nat @ N3 @ M2 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N3 ) @ M2 ) ) )
=> ( member @ nat @ ( gcd_Lcm @ nat @ M2 ) @ M2 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_4577_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_4578_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C2: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ) ).
% Gcd_mult
thf(fact_4579_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_4580_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A8: A,B6: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A8 @ B6 ) @ ( gcd_gcd @ A @ A8 @ B6 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_4581_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_4582_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A,B3: A] :
( ( finite_finite2 @ 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_4583_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A5: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A5
@ ( collect @ A
@ ^ [A8: A] : ( dvd_dvd @ A @ A8 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_4584_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_4585_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_4586_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit1 @ Y ) )
= ( times_times @ num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_4587_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_4588_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_4589_Lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,B2: A] :
( ( member @ A @ A6 @ A3 )
=> ( ( member @ A @ B2 @ A3 )
=> ( ( A6 != B2 )
=> ( algebr8660921524188924756oprime @ A @ A6 @ B2 ) ) ) )
=> ( ( gcd_Lcm @ A @ A3 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ A3 ) ) ) ) ) ) ) ).
% Lcm_coprime
thf(fact_4590_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
& ( algebr8660921524188924756oprime @ A @ B3 @ C2 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_4591_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A4 @ B3 ) )
= ( ( algebr8660921524188924756oprime @ A @ C2 @ A4 )
& ( algebr8660921524188924756oprime @ A @ C2 @ B3 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_4592_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_4593_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_4594_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_4595_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_4596_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_4597_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_4598_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_4599_coprime__crossproduct__nat,axiom,
! [A4: nat,D3: nat,B3: nat,C2: nat] :
( ( algebr8660921524188924756oprime @ nat @ A4 @ D3 )
=> ( ( algebr8660921524188924756oprime @ nat @ B3 @ C2 )
=> ( ( ( times_times @ nat @ A4 @ C2 )
= ( times_times @ nat @ B3 @ D3 ) )
= ( ( A4 = B3 )
& ( C2 = D3 ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_4600_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ A4 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_4601_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A4 ) ) ).
% coprime_1_left
thf(fact_4602_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_4603_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_4604_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_4605_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_4606_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ B3 @ C2 )
=> ( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% divides_mult
thf(fact_4607_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_4608_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_4609_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_4610_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_4611_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_4612_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_4613_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D3: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A4 )
=> ( ( dvd_dvd @ A @ E2 @ B3 )
=> ( dvd_dvd @ A @ E2 @ C2 ) ) ) )
=> ( ! [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_4614_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_4615_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% not_coprimeI
thf(fact_4616_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ~ ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A4 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_4617_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A8: A,B6: A] :
! [C5: A] :
( ( dvd_dvd @ A @ C5 @ A8 )
=> ( ( dvd_dvd @ A @ C5 @ B6 )
=> ( dvd_dvd @ A @ C5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_4618_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A4 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% coprimeI
thf(fact_4619_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N2: A,A4: A,M: A,B3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N2 @ A4 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ N2 @ B3 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N2 )
=> ( ( modulo_modulo @ A @ A4 @ M )
= ( modulo_modulo @ A @ B3 @ M ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_4620_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A4: A,N2: A,M: A,B3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ N2 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ B3 @ N2 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N2 )
=> ( ( modulo_modulo @ A @ A4 @ M )
= ( modulo_modulo @ A @ B3 @ M ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_4621_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( gcd_gcd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_4622_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( gcd_gcd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_4623_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( algebr8660921524188924756oprime @ A @ B3 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_4624_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( algebr8660921524188924756oprime @ A @ B3 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ B3 )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_4625_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_4626_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A8: A,B6: A] :
( ( gcd_gcd @ A @ A8 @ B6 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_4627_coprime__crossproduct__int,axiom,
! [A4: int,D3: int,B3: int,C2: int] :
( ( algebr8660921524188924756oprime @ int @ A4 @ D3 )
=> ( ( algebr8660921524188924756oprime @ int @ B3 @ C2 )
=> ( ( ( times_times @ int @ ( abs_abs @ int @ A4 ) @ ( abs_abs @ int @ C2 ) )
= ( times_times @ int @ ( abs_abs @ int @ B3 ) @ ( abs_abs @ int @ D3 ) ) )
= ( ( ( abs_abs @ int @ A4 )
= ( abs_abs @ int @ B3 ) )
& ( ( abs_abs @ int @ C2 )
= ( abs_abs @ int @ D3 ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_4628_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,D3: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B3 @ C2 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B3 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
& ( ( normal6383669964737779283malize @ A @ C2 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_4629_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_4630_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,A7: A,B4: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A4
= ( times_times @ A @ A7 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
=> ( ( B3
= ( times_times @ A @ B4 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A7 @ B4 ) ) ) ) ) ).
% gcd_coprime
thf(fact_4631_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ? [A19: A,B11: A] :
( ( A4
= ( times_times @ A @ A19 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
& ( B3
= ( times_times @ A @ B11 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
& ( algebr8660921524188924756oprime @ A @ A19 @ B11 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_4632_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_4633_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F2: A > nat,A3: set @ A,G: B > nat,B5: set @ B] :
( ( inj_on @ A @ nat @ F2 @ A3 )
=> ( ( inj_on @ B @ nat @ G @ B5 )
=> ( ! [A6: A,B2: B] :
( ( member @ A @ A6 @ A3 )
=> ( ( member @ B @ B2 @ B5 )
=> ( algebr8660921524188924756oprime @ nat @ ( F2 @ A6 ) @ ( G @ B2 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A8: A,B6: B] : ( times_times @ nat @ ( F2 @ A8 ) @ ( G @ B6 ) ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_4634_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_4635_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_4636_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_4637_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_4638_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_4639_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_4640_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_4641_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_4642_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_4643_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_4644_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_4645_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_4646_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_4647_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_4648_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_gcd_right
thf(fact_4649_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_4650_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_4651_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_lcm_right
thf(fact_4652_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_4653_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_4654_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_4655_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B3: A,D3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B3 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ( ( times_times @ A @ A4 @ D3 )
= ( times_times @ A @ B3 @ C2 ) )
= ( ( A4 = C2 )
& ( B3 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_4656_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_4657_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_4658_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F5: set @ ( C > ( set @ B ) ),G: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F5 ) @ G )
= ( bNF_collect @ A @ B
@ ( image2 @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F4: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F4 @ G )
@ F5 ) ) ) ).
% collect_comp
thf(fact_4659_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_4660_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_4661_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_4662_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_4663_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_4664_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_4665_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_less_of_rat_iff
thf(fact_4666_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_4667_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_le_of_rat_iff
thf(fact_4668_subset__mset_OacyclicI__order,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ B ),F2: B > ( multiset @ A )] :
( ! [A6: B,B2: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B2 ) @ R2 )
=> ( subset_mset @ A @ ( F2 @ B2 ) @ ( F2 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R2 ) ) ).
% subset_mset.acyclicI_order
thf(fact_4669_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_4670_collect__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_collect @ B @ A )
= ( ^ [F7: set @ ( B > ( set @ A ) ),X2: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( B > ( set @ A ) ) @ ( set @ A )
@ ^ [F4: B > ( set @ A )] : ( F4 @ X2 )
@ F7 ) ) ) ) ).
% collect_def
thf(fact_4671_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_4672_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_4673_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_4674_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_4675_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_4676_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_4677_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_4678_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_4679_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_4680_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_4681_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I: set @ B,F2: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I )
=> ( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I )
=> ( 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 @ I ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_4682_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,F2: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( subset_mset @ A @ ( F2 @ X3 ) @ ( G @ 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 ) ) @ G @ A3 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_4683_typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A,S: set @ B] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ S )
=> ( equiv_part_equivp @ B
@ ( bNF_eq_onp @ B
@ ^ [X2: B] : ( member @ B @ X2 @ S ) ) ) ) ).
% typedef_to_part_equivp
thf(fact_4684_open__typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A,P: B > $o] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( collect @ B @ P ) )
=> ( equiv_part_equivp @ B @ ( bNF_eq_onp @ B @ P ) ) ) ).
% open_typedef_to_part_equivp
thf(fact_4685_in__range_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ! [X3: nat] :
( ( member @ nat @ X3 @ As2 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_4686_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_4687_in__range_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( Y
= ( ~ ! [X2: nat] :
( ( member @ nat @ X2 @ As2 )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_4688_in__range_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ~ ! [X6: nat] :
( ( member @ nat @ X6 @ As2 )
=> ( ord_less @ nat @ X6 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_4689_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,X: A,H2: heap_ext @ product_unit,As: set @ nat] :
( ( sngr_assn_raw @ A @ R2 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ( ( ref_get @ A @ H2 @ R2 )
= X )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ R2 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ R2 ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% sngr_assn_raw.simps
thf(fact_4690_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( Y
= ( ~ ( ( ( ref_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
thf(fact_4691_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ~ ( ( ( ref_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% sngr_assn_raw.elims(2)
thf(fact_4692_relH__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,R2: ref @ A] :
( ( relH @ As @ H2 @ H3 )
=> ( ( member @ nat @ ( addr_of_ref @ A @ R2 ) @ As )
=> ( ( ref_get @ A @ H2 @ R2 )
= ( ref_get @ A @ H3 @ R2 ) ) ) ) ) ).
% relH_ref
thf(fact_4693_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( ( ref_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% sngr_assn_raw.elims(3)
thf(fact_4694_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 ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( 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 ) @ H @ As2 ) ) ) )
=> ( ( ( ref_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
thf(fact_4695_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 ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( 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 ) @ H @ As2 ) ) ) )
=> ~ ( ( ( ref_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
thf(fact_4696_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 ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( Y
= ( ( ( ref_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H ) ) ) )
=> ~ ( 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 ) @ H @ As2 ) ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
thf(fact_4697_relH__set__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,As: set @ nat,H2: heap_ext @ product_unit,X: A] :
( ~ ( member @ nat @ ( addr_of_ref @ A @ R2 ) @ As )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( relH @ As @ H2 @ ( ref_set @ A @ R2 @ X @ H2 ) ) ) ) ) ).
% relH_set_ref
thf(fact_4698_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: array @ A,X: list @ A,H2: heap_ext @ product_unit,As: set @ nat] :
( ( snga_assn_raw @ A @ R2 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
= ( ( ( array_get @ A @ H2 @ R2 )
= X )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ R2 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ R2 ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% snga_assn_raw.simps
thf(fact_4699_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( Y
= ( ~ ( ( ( array_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
thf(fact_4700_relH__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As: set @ nat,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,R2: array @ A] :
( ( relH @ As @ H2 @ H3 )
=> ( ( member @ nat @ ( addr_of_array @ A @ R2 ) @ As )
=> ( ( array_get @ A @ H2 @ R2 )
= ( array_get @ A @ H3 @ R2 ) ) ) ) ) ).
% relH_array
thf(fact_4701_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( ( array_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% snga_assn_raw.elims(3)
thf(fact_4702_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ~ ( ( ( array_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% snga_assn_raw.elims(2)
thf(fact_4703_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 ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( 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 ) @ H @ As2 ) ) ) )
=> ( ( ( array_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
thf(fact_4704_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 ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( 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 ) @ H @ As2 ) ) ) )
=> ~ ( ( ( array_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
thf(fact_4705_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 ) ) )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( Y
= ( ( ( array_get @ A @ H @ X )
= Xa )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H ) ) ) )
=> ~ ( 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 ) @ H @ As2 ) ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
thf(fact_4706_relH__set__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: array @ A,As: set @ nat,H2: heap_ext @ product_unit,X: list @ A] :
( ~ ( member @ nat @ ( addr_of_array @ A @ R2 ) @ As )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( relH @ As @ H2 @ ( array_set @ A @ R2 @ X @ H2 ) ) ) ) ) ).
% relH_set_array
thf(fact_4707_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As8: A > B] :
! [I4: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As8 @ I4 ) @ ( As8 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_4708_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_4709_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A5: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R5 ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A5 )
& ( X2 != Y3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_4710_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B5: set @ A,A3: set @ B] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F4: B > A] :
( ( image2 @ B @ A @ F4 @ A3 )
= B5 ) )
= ( 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 @ B5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_4711_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A5: set @ A,F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A8: A] :
( ( Uu
= ( product_Pair @ A @ B @ A8 @ ( F4 @ A8 ) ) )
& ( member @ A @ A8 @ A5 ) ) ) ) ) ).
% Gr_def
thf(fact_4712_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_4713_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_4714_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_4715_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ B5 )
= 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 @ B5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_4716_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_4717_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_4718_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_4719_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_4720_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_4721_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_4722_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_4723_exists__minim__Well__order,axiom,
! [A: $tType,R4: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R4
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R4 )
=> ( order_well_order_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R4 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_4724_card__of__Times2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: 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 @ B5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_4725_card__of__Times1,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: 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 @ B5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu: B] : A3 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_4726_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set @ A,B5: 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 @ B5 ) ) @ ( 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 @ B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_4727_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( 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] : B5 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_4728_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( 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 @ B5
@ ^ [Uu: B] : A3 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_4729_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_4730_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_4731_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set @ A,B1: B,B22: B,B5: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) )
=> ( ( ( B1 != B22 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B1 @ ( insert2 @ B @ B22 @ ( bot_bot @ ( set @ B ) ) ) ) @ B5 ) )
=> ( 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 @ B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_4732_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ( ( sum_Plus @ A @ B @ A3 @ B5 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_4733_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( 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 @ B5
@ ^ [Uu: B] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_4734_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( 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] : B5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_4735_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_4736_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_4737_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_4738_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( 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 @ B5 ) @ ( 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] : B5 ) )
@ ( 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 @ B5
@ ^ [Uu: B] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_4739_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P3: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ B @ P3 )
!= ( 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 ) ) @ P3 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R2 )
@ ^ [Uu: A] : ( field2 @ B @ P3 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( 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 @ P3 )
@ ^ [Uu: B] : ( field2 @ A @ R2 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_4740_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( 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 ) ) ) @ R2
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu: B] : ( field2 @ A @ R2 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_4741_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),B5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( B5
!= ( 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 ) ) ) @ R2
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R2 )
@ ^ [Uu: A] : B5 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_4742_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert2 @ $o @ $false @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_4743_Card__order__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( Y != Z2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( ( X != Z2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_4744_Card__order__wo__rel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( bNF_Wellorder_wo_rel @ A @ R2 ) ) ).
% Card_order_wo_rel
thf(fact_4745_infinite__Card__order__limit,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( A4 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ R2 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_4746_exists__minim__Card__order,axiom,
! [A: $tType,R4: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R4
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R4 )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R4 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_4747_Card__order__empty,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_4748_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),B3: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_4749_card__of__empty1,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_4750_Cinfinite__limit__finite,axiom,
! [A: $tType,X4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ X4 )
=> ( ( Xa2 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_4751_czeroI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_4752_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_4753_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_4754_Cinfinite__limit,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( X != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X3 ) @ R2 ) ) ) ) ).
% Cinfinite_limit
thf(fact_4755_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R2: set @ ( product_prod @ A @ A ),X22: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ X22 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ( X1 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X3 ) @ R2 )
& ( X22 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X3 ) @ R2 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_4756_czeroE,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_4757_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_4758_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P24: 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 ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( ( field2 @ A @ P24 )
= ( 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 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_4759_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_4760_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_4761_Rep__unit__inject,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( product_Rep_unit @ X )
= ( product_Rep_unit @ Y ) )
= ( X = Y ) ) ).
% Rep_unit_inject
thf(fact_4762_Rep__unit__inverse,axiom,
! [X: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
= X ) ).
% Rep_unit_inverse
thf(fact_4763_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_4764_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_4765_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P14: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: 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 ) ) @ P14 @ 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 ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P24 )
= ( 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 @ P14 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_4766_Rep__unit,axiom,
! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_4767_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X
= ( product_Abs_unit @ Y2 ) )
=> ~ ( member @ $o @ Y2 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_4768_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_4769_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y2: $o] :
( ( member @ $o @ Y2 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P @ ( product_Abs_unit @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_4770_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T6: set @ C,H2: B > C,S: set @ B,T3: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) @ ( minus_minus @ ( set @ C ) @ T3 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B2: C] :
( ( member @ C @ B2 @ T6 )
=> ( ( G @ B2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T3 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4771_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_4772_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_finite2 @ 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_4773_bijI_H,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ! [X3: A,Y2: A] :
( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
= ( X3 = Y2 ) )
=> ( ! [Y2: B] :
? [X6: A] :
( Y2
= ( F2 @ X6 ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI'
thf(fact_4774_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X4: set @ A,F2: A > B,Y5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ X4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
= ( X3 = Y2 ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X4 )
=> ( member @ B @ ( F2 @ X3 ) @ Y5 ) )
=> ( ! [Y2: B] :
( ( member @ B @ Y2 @ Y5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ X4 )
& ( Y2
= ( F2 @ X6 ) ) ) )
=> ( bij_betw @ A @ B @ F2 @ X4 @ Y5 ) ) ) ) ).
% bij_betwI'
thf(fact_4775_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_4776_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_4777_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_4778_folding__on_Oinsert,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_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_4779_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_4780_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A3: set @ A,F2: A > B,A10: set @ B] :
( ~ ( member @ A @ B3 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B3 ) @ A10 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A10 )
= ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A10 @ ( insert2 @ B @ ( F2 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_4781_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A3: set @ A,F2: A > B,A10: set @ B] :
( ~ ( member @ A @ B3 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B3 ) @ A10 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A10 )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A10 @ ( insert2 @ B @ ( F2 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_4782_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B5: set @ B,C6: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ B5 )
=> ( ( bij_betw @ A @ B @ F2 @ C6 @ D4 )
=> ( ( ( inf_inf @ ( set @ B ) @ B5 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C6 ) @ ( sup_sup @ ( set @ B ) @ B5 @ D4 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_4783_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C6: set @ A,B5: set @ B,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C6 ) @ ( sup_sup @ ( set @ B ) @ B5 @ D4 ) )
=> ( ( bij_betw @ A @ B @ F2 @ C6 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ C6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B5 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ A3 @ B5 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_4784_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: 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 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F: B > A,B7: set @ B] : ( bij_betw @ B @ A @ F @ B7 @ A3 ) ) ).
% ex_bij_betw
thf(fact_4785_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C6: set @ B,G: A > B,B5: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ C6 )
=> ( ( bij_betw @ A @ B @ G @ B5 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C6 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A3 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B5 )
@ ( sup_sup @ ( set @ B ) @ C6 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_4786_infinite__imp__bij__betw2,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H: A > A] : ( bij_betw @ A @ A @ H @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_4787_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H: A > A] : ( bij_betw @ A @ A @ H @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_4788_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_finite2 @ 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_4789_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F2: A > B > B,X: A,A3: set @ A,Z2: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_4790_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_4791_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ) ).
% trancl_def
thf(fact_4792_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: ( 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 ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A6: A,B2: B] :
( ( R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( P @ A6 @ B2 ) )
=> ( ! [A6: A,B2: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( ( R2 @ ( product_Pair @ A @ B @ A6 @ B2 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A6 @ B2 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_4793_tranclp__trancl__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_4794_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R5: A > A > $o,A8: A,B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_4795_reflp__refl__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 ) ) ).
% reflp_refl_eq
thf(fact_4796_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ A ),X2: B] :
( fChoice @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) @ R3 ) ) ) ) ).
% fun_of_rel_def
thf(fact_4797_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X11: A,Y9: B] :
( ( X = X11 )
& ( Y = Y9 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_4798_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_4799_reflp__inf,axiom,
! [A: $tType,R2: A > A > $o,S2: A > A > $o] :
( ( reflp @ A @ R2 )
=> ( ( reflp @ A @ S2 )
=> ( reflp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S2 ) ) ) ) ).
% reflp_inf
thf(fact_4800_reflpD,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( reflp @ A @ R2 )
=> ( R2 @ X @ X ) ) ).
% reflpD
thf(fact_4801_reflpE,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( reflp @ A @ R2 )
=> ( R2 @ X @ X ) ) ).
% reflpE
thf(fact_4802_reflpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A] : ( R2 @ X3 @ X3 )
=> ( reflp @ A @ R2 ) ) ).
% reflpI
thf(fact_4803_reflp__def,axiom,
! [A: $tType] :
( ( reflp @ A )
= ( ^ [R5: A > A > $o] :
! [X2: A] : ( R5 @ X2 @ X2 ) ) ) ).
% reflp_def
thf(fact_4804_reflp__mono,axiom,
! [A: $tType,R4: A > A > $o,Q2: A > A > $o] :
( ( reflp @ A @ R4 )
=> ( ! [X3: A,Y2: A] :
( ( R4 @ X3 @ Y2 )
=> ( Q2 @ X3 @ Y2 ) )
=> ( reflp @ A @ Q2 ) ) ) ).
% reflp_mono
thf(fact_4805_reflp__equality,axiom,
! [A: $tType] :
( reflp @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) ).
% reflp_equality
thf(fact_4806_reflp__sup,axiom,
! [A: $tType,R2: A > A > $o,S2: A > A > $o] :
( ( reflp @ A @ R2 )
=> ( ( reflp @ A @ S2 )
=> ( reflp @ A @ ( sup_sup @ ( A > A > $o ) @ R2 @ S2 ) ) ) ) ).
% reflp_sup
thf(fact_4807_reflp__eq,axiom,
! [A: $tType] :
( ( reflp @ A )
= ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) ) ).
% reflp_eq
thf(fact_4808_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_4809_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_4810_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F4: A > B,A5: set @ A,X2: B] :
( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A5 )
& ( ( F4 @ Y3 )
= X2 ) ) ) ) ) ).
% inv_on_def
thf(fact_4811_pick__middlep__def,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( bNF_pick_middlep @ B @ A @ C )
= ( ^ [P2: B > A > $o,Q: A > C > $o,A8: B,C5: C] :
( fChoice @ A
@ ^ [B6: A] :
( ( P2 @ A8 @ B6 )
& ( Q @ B6 @ C5 ) ) ) ) ) ).
% pick_middlep_def
thf(fact_4812_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P2: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B6: B] : ( P2 @ ( product_Pair @ A @ B @ A8 @ B6 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_4813_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A12: A,X_12: B] : ( P @ A12 @ X_12 )
=> ( ! [B19: B,B25: B] :
( ? [A6: A] : ( P @ A6 @ B19 )
=> ( ? [A6: A] : ( P @ A6 @ B25 )
=> ( B19 = B25 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A8: A] :
? [X7: B] : ( P @ A8 @ X7 ) )
@ ( the @ B
@ ^ [B6: B] :
? [A8: A] : ( P @ A8 @ B6 ) ) ) ) ) ).
% some_theI
thf(fact_4814_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_4815_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ).
% inj_on_fst_map_to_set
thf(fact_4816_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F26: T,X2: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F26 @ X2 ) ) ) ) ).
% old.rec_bool_def
thf(fact_4817_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ^ [B6: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B6 ) ) ) ) ).
% sum_encode_def
thf(fact_4818_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_4819_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_4820_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V )
@ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [V5: B] :
( Uu
= ( product_Pair @ A @ B @ K @ V5 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_4821_set__to__map__inverse,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S )
=> ( ( map_to_set @ A @ B @ ( set_to_map @ A @ B @ S ) )
= S ) ) ).
% set_to_map_inverse
thf(fact_4822_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M4: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M4 ) ) ) ) ).
% map_to_set_ran
thf(fact_4823_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K2: set @ A,M: A > ( option @ B ),V: B] :
( ( member @ A @ K @ K2 )
=> ( ( map_mmupd @ A @ B @ M @ K2 @ V @ K )
= ( some @ B @ V ) ) ) ).
% mmupd_in_upd
thf(fact_4824_map__mmupd__def,axiom,
! [A: $tType,B: $tType] :
( ( map_mmupd @ B @ A )
= ( ^ [M4: B > ( option @ A ),K6: set @ B,V4: A,K5: B] : ( if @ ( option @ A ) @ ( member @ B @ K5 @ K6 ) @ ( some @ A @ V4 ) @ ( M4 @ K5 ) ) ) ) ).
% map_mmupd_def
thf(fact_4825_map__mmupdE,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),K2: set @ B,V: A,K: B,X: A] :
( ( ( map_mmupd @ B @ A @ M @ K2 @ V @ K )
= ( some @ A @ X ) )
=> ( ( ~ ( member @ B @ K @ K2 )
=> ( ( M @ K )
!= ( some @ A @ X ) ) )
=> ~ ( ( member @ B @ K @ K2 )
=> ( X != V ) ) ) ) ).
% map_mmupdE
thf(fact_4826_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X )
=> ~ ! [M7: A] :
( ( X
= ( some @ A @ M7 ) )
=> ~ ( ord_less_eq @ A @ M @ M7 ) ) ) ) ).
% le_some_optE
thf(fact_4827_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_4828_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( set_to_map @ B @ A @ S ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S ) ) ).
% set_to_map_ran
thf(fact_4829_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M ) )
= M ) ).
% map_to_set_inverse
thf(fact_4830_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S: set @ ( product_prod @ A @ B )] :
( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ( ( set_to_map @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ Kv @ S ) )
= ( fun_upd @ A @ ( option @ B ) @ ( set_to_map @ A @ B @ S ) @ ( product_fst @ A @ B @ Kv ) @ ( some @ B @ ( product_snd @ A @ B @ Kv ) ) ) ) ) ).
% set_to_map_insert
thf(fact_4831_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M4: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K5: A,V4: B] :
( ( M4 @ K5 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_4832_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_4833_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M4: A > ( option @ B ),P2: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K5: A,V4: B] :
( ( ( M4 @ K5 )
= ( some @ B @ V4 ) )
& ( P2 @ ( product_Pair @ A @ B @ K5 @ V4 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_4834_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_4835_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_4836_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S6: set @ ( product_prod @ B @ A ),K5: B] :
( eps_Opt @ A
@ ^ [V4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K5 @ V4 ) @ S6 ) ) ) ) ).
% set_to_map_def
thf(fact_4837_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A,X: A] :
( ( the_default @ A @ Uu2 @ ( some @ A @ X ) )
= X ) ).
% the_default.simps(1)
thf(fact_4838_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ( ^ [Y4: A,Z5: A] : Y4 = Z5
@ X ) )
= ( some @ A @ X ) ) ).
% some_opt_sym_eq_trivial
thf(fact_4839_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_4840_some__opt__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ^ [Y3: A] : Y3 = X )
= ( some @ A @ X ) ) ).
% some_opt_eq_trivial
thf(fact_4841_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: A > $o,X: A] :
( ! [X10: A] :
( ( P @ X )
=> ( ( P @ X10 )
=> ( X10 = X ) ) )
=> ( ( ( eps_Opt @ A @ P )
= ( some @ A @ X ) )
= ( P @ X ) ) ) ).
% Eps_Opt_eq_Some
thf(fact_4842_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( ( eps_Opt @ A @ P )
= ( some @ A @ X ) )
=> ( P @ X ) ) ).
% Eps_Opt_eq_Some_implies
thf(fact_4843_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P2: A > $o] :
( if @ ( option @ A )
@ ? [X7: A] : ( P2 @ X7 )
@ ( some @ A @ ( fChoice @ A @ P2 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_4844_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S6: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S6
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S6 ) ) ) ) ).
% dflt_None_set_def
thf(fact_4845_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X: B,Y: A,Z2: A] :
( ( ( M @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_4846_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X2: A,Y3: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_4847_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_4848_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_4849_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_4850_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_4851_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X7: A] : ( P @ X7 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_4852_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_4853_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X2: A] : ( if @ ( option @ B ) @ ( X2 = K ) @ ( some @ B @ V ) @ ( M @ X2 ) ) )
= ( insert2 @ A @ K @ ( dom @ A @ B @ M ) ) ) ).
% map_update_eta_repair(1)
thf(fact_4854_dom__const_H,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( dom @ A @ B
@ ^ [X2: A] : ( some @ B @ ( F2 @ X2 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const'
thf(fact_4855_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_4856_dom__mmupd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K2: set @ A,V: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M @ K2 @ V ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M ) @ K2 ) ) ).
% dom_mmupd
thf(fact_4857_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_4858_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_4859_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( ( M @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X2: A] : ( if @ ( option @ B ) @ ( X2 = K ) @ ( some @ B @ V ) @ ( M @ X2 ) ) )
= ( insert2 @ B @ V @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_4860_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_4861_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_4862_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A4: B,B3: A] :
( ( ( M @ A4 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A4 @ ( some @ A @ B3 ) ) )
= ( insert2 @ A @ B3 @ ( ran @ B @ A @ M ) ) ) ) ).
% ran_map_upd
thf(fact_4863_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_4864_insert__dom,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,Y: A] :
( ( ( F2 @ X )
= ( some @ A @ Y ) )
=> ( ( insert2 @ B @ X @ ( dom @ B @ A @ F2 ) )
= ( dom @ B @ A @ F2 ) ) ) ).
% insert_dom
thf(fact_4865_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 ) ) ) )
= ( ? [V4: B] :
( F2
= ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( some @ B @ V4 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_4866_dom__minus,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,A3: set @ B] :
( ( ( F2 @ X )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ ( insert2 @ B @ X @ A3 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ A3 ) ) ) ).
% dom_minus
thf(fact_4867_nempty__dom,axiom,
! [B: $tType,A: $tType,E4: A > ( option @ B )] :
( ( E4
!= ( ^ [X2: A] : ( none @ B ) ) )
=> ~ ! [M3: A] :
~ ( member @ A @ M3 @ ( dom @ A @ B @ E4 ) ) ) ).
% nempty_dom
thf(fact_4868_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M ) @ ( dom @ A @ B @ M8 ) ) ) ) ).
% le_map_dom_mono
thf(fact_4869_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_4870_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ M2 ) ) ) ).
% map_dom_ran_finite
thf(fact_4871_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( the_default @ A @ X @ ( none @ A ) )
= X ) ).
% the_default.simps(2)
thf(fact_4872_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M ) )
= ( M
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_4873_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_4874_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_4875_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: A] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( ( finite_card @ A @ ( dom @ A @ B @ M2 ) )
= ( finite_card @ B @ ( ran @ A @ B @ M2 ) ) )
=> ( ( member @ A @ X @ ( dom @ A @ B @ M2 ) )
=> ( ( ( M2 @ X )
= ( M2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% map_card_eq_iff
thf(fact_4876_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_4877_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M ) ) ) ).
% finite_map_to_set
thf(fact_4878_map__to__set__dom,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M4: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M4 ) ) ) ) ).
% map_to_set_dom
thf(fact_4879_set__to__map__dom,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( dom @ A @ B @ ( set_to_map @ A @ B @ S ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ).
% set_to_map_dom
thf(fact_4880_card__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M ) )
= ( finite_card @ A @ ( dom @ A @ B @ M ) ) ) ).
% card_map_to_set
thf(fact_4881_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_4882_Sup__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) )
= ( ^ [A5: set @ ( option @ A )] :
( if @ ( option @ A )
@ ( ( A5
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( A5
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
@ ( none @ A )
@ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A5 ) ) ) ) ) ) ) ).
% Sup_option_def
thf(fact_4883_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_4884_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= F2 ) ).
% restrict_map_UNIV
thf(fact_4885_restrict__map__self,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( dom @ A @ B @ M ) )
= M ) ).
% restrict_map_self
thf(fact_4886_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_4887_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_4888_restrict__map__inv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_inv
thf(fact_4889_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D4: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_4890_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D4 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D4 )
= ( restrict_map @ A @ B @ M @ D4 ) ) ) ) ).
% restrict_fun_upd
thf(fact_4891_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D4 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_4892_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A3 ) ) )
=> ( ( M @ K )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_4893_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A3 ) ) )
=> ( member @ A @ K @ A3 ) ) ).
% graph_restrictD(1)
thf(fact_4894_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),X4: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M @ X4 ) @ M ) ) ).
% le_map_restrict
thf(fact_4895_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),R4: set @ A,M8: A > ( option @ B ),R6: set @ A] :
( ( ( restrict_map @ A @ B @ M @ R4 )
= M8 )
=> ( ( ord_less_eq @ ( set @ A ) @ R6 @ R4 )
=> ( ( restrict_map @ A @ B @ M @ R6 )
= ( restrict_map @ A @ B @ M8 @ R6 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_4896_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: set @ B,K: B,V: A] :
( ( ( restrict_map @ B @ A @ M @ A3 @ K )
= ( some @ A @ V ) )
= ( ( ( M @ K )
= ( some @ A @ V ) )
& ( member @ B @ K @ A3 ) ) ) ).
% restrict_map_eq(2)
thf(fact_4897_restrict__map__insert,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),A4: A,A3: set @ A] :
( ( restrict_map @ A @ B @ F2 @ ( insert2 @ A @ A4 @ A3 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F2 @ A3 ) @ A4 @ ( F2 @ A4 ) ) ) ).
% restrict_map_insert
thf(fact_4898_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_4899_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V: A] :
( ( ( M @ K )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_4900_graph__ranD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( graph @ A @ B @ M ) )
=> ( member @ B @ ( product_snd @ A @ B @ X ) @ ( ran @ A @ B @ M ) ) ) ).
% graph_ranD
thf(fact_4901_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),X: B,S2: set @ B] :
( ( ( M @ X )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ ( insert2 @ B @ X @ S2 ) ) )
= ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ S2 ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_4902_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: set @ B,K: B] :
( ( ( restrict_map @ B @ A @ M @ A3 @ K )
= ( none @ A ) )
= ( ~ ( member @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M ) @ A3 ) ) ) ) ).
% restrict_map_eq(1)
thf(fact_4903_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),S: set @ A,K: A,V: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F2 @ S ) @ K @ ( some @ B @ V ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ K @ ( some @ B @ V ) ) @ ( insert2 @ A @ K @ S ) ) ) ).
% restrict_map_upd
thf(fact_4904_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M4: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A8: A,B6: B] :
( ( Uu
= ( product_Pair @ A @ B @ A8 @ B6 ) )
& ( ( M4 @ A8 )
= ( some @ B @ B6 ) ) ) ) ) ) ).
% graph_def
thf(fact_4905_snd__graph__ran,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( graph @ B @ A @ M ) )
= ( ran @ B @ A @ M ) ) ).
% snd_graph_ran
thf(fact_4906_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),D4: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_4907_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_4908_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_4909_these__insert__Some,axiom,
! [A: $tType,X: A,A3: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X ) @ A3 ) )
= ( insert2 @ A @ X @ ( these @ A @ A3 ) ) ) ).
% these_insert_Some
thf(fact_4910_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_4911_these__not__empty__eq,axiom,
! [A: $tType,B5: set @ ( option @ A )] :
( ( ( these @ A @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B5
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B5
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_4912_these__empty__eq,axiom,
! [A: $tType,B5: set @ ( option @ A )] :
( ( ( these @ A @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B5
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B5
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_4913_ran__map__add,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M12 ) @ ( dom @ A @ B @ M23 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M12 @ M23 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M12 ) @ ( ran @ A @ B @ M23 ) ) ) ) ).
% ran_map_add
thf(fact_4914_ran__add,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( dom @ A @ B @ G ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F2 @ G ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F2 ) @ ( ran @ A @ B @ G ) ) ) ) ).
% ran_add
thf(fact_4915_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: ( A > ( option @ B ) ) > A > ( option @ B ),A3: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ ( dom @ A @ B @ ( F2 @ A3 ) ) )
=> ( ( A3
= ( restrict_map @ A @ B @ ( F2 @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) )
= ( ( map_le @ A @ B @ A3 @ ( F2 @ A3 ) )
& ( S2
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F2 @ A3 ) ) @ ( dom @ A @ B @ A3 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_4916_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K2: set @ A,V: B] : ( map_le @ A @ B @ M @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M ) ) @ V ) ) ).
% le_map_mmupd_not_dom
thf(fact_4917_map__leI,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ! [X3: A,V3: B] :
( ( ( M12 @ X3 )
= ( some @ B @ V3 ) )
=> ( ( M23 @ X3 )
= ( some @ B @ V3 ) ) )
=> ( map_le @ A @ B @ M12 @ M23 ) ) ).
% map_leI
thf(fact_4918_map__leD,axiom,
! [A: $tType,B: $tType,M12: A > ( option @ B ),M23: A > ( option @ B ),K: A,V: B] :
( ( map_le @ A @ B @ M12 @ M23 )
=> ( ( ( M12 @ K )
= ( some @ B @ V ) )
=> ( ( M23 @ K )
= ( some @ B @ V ) ) ) ) ).
% map_leD
thf(fact_4919_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B ),N2: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N2 ) @ ( map_add @ A @ B @ M8 @ N2 ) ) ) ) ).
% map_add_first_le
thf(fact_4920_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: B > ( option @ A ),K: B,F2: B > ( option @ A )] :
( ( ( G @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F2 @ G @ K )
= ( F2 @ K ) ) ) ).
% map_add_find_left
thf(fact_4921_map__add__left__None,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),K: B,G: B > ( option @ A )] :
( ( ( F2 @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F2 @ G @ K )
= ( G @ K ) ) ) ).
% map_add_left_None
thf(fact_4922_map__add__comm,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M12 ) @ ( dom @ A @ B @ M23 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M12 @ M23 )
= ( map_add @ A @ B @ M23 @ M12 ) ) ) ).
% map_add_comm
thf(fact_4923_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A3: A > ( option @ B ),B5: A > ( option @ B ),C6: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A3 ) @ ( dom @ A @ B @ B5 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A3 @ ( map_add @ A @ B @ B5 @ C6 ) )
= ( map_add @ A @ B @ B5 @ ( map_add @ A @ B @ A3 @ C6 ) ) ) ) ).
% map_add_left_comm
thf(fact_4924_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B ),N2: A > ( option @ B ),N8: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N2 @ N8 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M8 ) @ ( dom @ A @ B @ N8 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N2 ) @ ( map_add @ A @ B @ M8 @ N8 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_4925_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K2: set @ A,K8: set @ A,M: A > ( option @ B ),V: B] :
( ( ord_less_eq @ ( set @ A ) @ K2 @ K8 )
=> ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M ) ) @ V ) @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K8 @ ( dom @ A @ B @ M ) ) @ V ) ) ) ).
% map_mmupd_update_less
thf(fact_4926_graph__map__add,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M12 ) @ ( dom @ A @ B @ M23 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M12 @ M23 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M12 ) @ ( graph @ A @ B @ M23 ) ) ) ) ).
% graph_map_add
thf(fact_4927_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: ( A > ( option @ B ) ) > A > ( option @ B ),A3: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( dom @ A @ B @ ( F2 @ A3 ) ) )
& ( A3
= ( restrict_map @ A @ B @ ( F2 @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) ) )
= ( ( map_le @ A @ B @ A3 @ ( F2 @ A3 ) )
& ( S2
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F2 @ A3 ) ) @ ( dom @ A @ B @ A3 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_4928_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M4: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( the2 @ B @ ( M4 @ X2 ) ) )
@ ( dom @ A @ B @ M4 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_4929_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D4: set @ A,M: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ D4 @ ( dom @ A @ B @ M ) )
=> ( ( inj_on @ A @ ( option @ B ) @ M @ D4 )
=> ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M ) @ D4 ) ) ) ).
% inj_on_map_the
thf(fact_4930_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_4931_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_4932_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_4933_ran__is__image,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M5: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M5 ) @ ( dom @ B @ A @ M5 ) ) ) ) ).
% ran_is_image
thf(fact_4934_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_4935_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_4936_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_4937_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_4938_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F4: A > A > A,Z3: A,Inverse2: A > A] :
( ! [A8: A] :
( ( F4 @ Z3 @ A8 )
= A8 )
& ! [A8: A] :
( ( F4 @ ( Inverse2 @ A8 ) @ A8 )
= Z3 ) ) ) ) ).
% group_axioms_def
thf(fact_4939_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_4940_ordering__top__axioms__def,axiom,
! [A: $tType] :
( ( ordering_top_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Top2: A] :
! [A8: A] : ( Less_eq2 @ A8 @ Top2 ) ) ) ).
% ordering_top_axioms_def
thf(fact_4941_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs ) )
= ( minus_minus @ nat @ To @ From ) ) ) ) ).
% slice_len
thf(fact_4942_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R4: set @ ( product_prod @ A @ A ),N2: nat,M: nat] :
( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N3 ) ) @ ( nth @ A @ Xs @ N3 ) ) @ R4 ) )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ ( nth @ A @ Xs @ M ) ) @ ( transitive_trancl @ A @ R4 ) ) ) ) ) ).
% nth_step_trancl
thf(fact_4943_slice__eq__bounds__empty,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( slice @ A @ I2 @ I2 @ Xs )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_4944_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_4945_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_4946_length__ge__1__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
= ( L
!= ( nil @ A ) ) ) ).
% length_ge_1_conv
thf(fact_4947_obtain__list__from__elements,axiom,
! [A: $tType,N2: nat,P: A > nat > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ? [Li: A] : ( P @ Li @ I3 ) )
=> ~ ! [L3: list @ A] :
( ( ( size_size @ ( list @ A ) @ L3 )
= N2 )
=> ~ ! [I8: nat] :
( ( ord_less @ nat @ I8 @ N2 )
=> ( P @ ( nth @ A @ L3 @ I8 ) @ I8 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_4948_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_4949_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_4950_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I2: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I2 ) ) ) ) ) ) ).
% slice_nth
thf(fact_4951_product__nth,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_4952_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( compow @ ( A > A ) @ N4 @ ( times_times @ A @ A8 ) @ ( F4 @ ( nth @ B @ Xs3 @ N4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4953_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs3 @ N4 ) ) @ ( power_power @ A @ A8 @ N4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4954_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_4955_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: nat,J2: nat,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I2 )
=> ( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I2 ) @ X )
=> ( ( ord_less @ A @ X @ ( nth @ A @ L @ J2 ) )
=> ~ ! [K3: nat] :
( ( ord_less_eq @ nat @ I2 @ K3 )
=> ( ( ord_less @ nat @ K3 @ J2 )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K3 ) @ X )
=> ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_4956_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A1: list @ A,P: ( A > A > $o ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [R8: A > A > $o,Xs4: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs4 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs4 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs4 ) ) ) )
=> ( P @ R8 @ Xs4 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_4957_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_4958_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R4: A > A > $o] :
( ( mergesort_by_rel @ A @ R4 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% mergesort_by_rel_simps(1)
thf(fact_4959_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R4: A > A > $o,Ys: list @ A] :
( ( merges9089515139780605204_merge @ A @ R4 @ ( nil @ A ) @ Ys )
= Ys ) ).
% mergesort_by_rel_merge_simps(3)
thf(fact_4960_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R4: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X3: A,Y2: A] :
( ( R4 @ X3 @ Y2 )
| ( R4 @ Y2 @ X3 ) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( ( R4 @ X3 @ Y2 )
=> ( ( R4 @ Y2 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( ( sorted_wrt @ A @ R4 @ ( merges9089515139780605204_merge @ A @ R4 @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ R4 @ Xs )
& ( sorted_wrt @ A @ R4 @ Ys ) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
thf(fact_4961_sorted__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_mergesort_by_rel
thf(fact_4962_sorted__wrt__mergesort__by__rel,axiom,
! [X14: $tType,R4: X14 > X14 > $o,Xs: list @ X14] :
( ! [X3: X14,Y2: X14] :
( ( R4 @ X3 @ Y2 )
| ( R4 @ Y2 @ X3 ) )
=> ( ! [X3: X14,Y2: X14,Z4: X14] :
( ( R4 @ X3 @ Y2 )
=> ( ( R4 @ Y2 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( sorted_wrt @ X14 @ R4 @ ( mergesort_by_rel @ X14 @ R4 @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_4963_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R4: A > A > $o,Xs: list @ A] :
( ( merges9089515139780605204_merge @ A @ R4 @ Xs @ ( nil @ A ) )
= Xs ) ).
% mergesort_by_rel_merge_simps(2)
thf(fact_4964_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R4: 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 ) @ R4 @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R4 @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R4 @ Xs )
= ( merges9089515139780605204_merge @ A @ R4 @ ( mergesort_by_rel @ A @ R4 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R4 @ ( 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_4965_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_4966_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R3: 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 @ R3 @ ( mergesort_by_rel @ A @ R3 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R3 @ ( 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_4967_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_4968_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R4: A > A > $o,X1: A,X22: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R4 @ ( 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 @ R4 @ ( mergesort_by_rel @ A @ R4 @ Xs12 ) @ ( mergesort_by_rel @ A @ R4 @ 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_4969_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_4970_mergesort__by__rel__permutes,axiom,
! [A: $tType,R4: A > A > $o,Xs: list @ A] :
( ( mset @ A @ ( mergesort_by_rel @ A @ R4 @ Xs ) )
= ( mset @ A @ Xs ) ) ).
% mergesort_by_rel_permutes
thf(fact_4971_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R4: A > A > $o,X: A] :
( ( mergesort_by_rel @ A @ R4 @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_4972_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R4: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R4 @ Xs @ Ys ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys ) ) ) ).
% mset_mergesort_by_rel_merge
thf(fact_4973_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_4974_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R4: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ( R4 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R4 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R4 @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ) )
& ( ~ ( R4 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R4 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R4 @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_4975_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R4: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs4: list @ A] : ( P @ Xs4 @ ( nil @ B ) )
=> ( ! [X_1: list @ B] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X3: A,Xs4: list @ A,Y2: B,Ys2: list @ B] :
( ( R4 @ X3 @ Y2 )
=> ( ( P @ Xs4 @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( P @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( ! [X3: A,Xs4: list @ A,Y2: B,Ys2: list @ B] :
( ~ ( R4 @ X3 @ Y2 )
=> ( ( P @ ( cons @ A @ X3 @ Xs4 ) @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_4976_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X12: A,X23: A,Xs4: list @ A] :
( ( P @ Xs4 )
=> ( P @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs4 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_4977_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E2: A,W12: list @ A,W23: list @ B] :
( ( P @ W12 @ W23 )
=> ( P @ ( cons @ A @ E2 @ W12 ) @ W23 ) )
=> ( ! [E2: B,W13: list @ A,W24: list @ B] :
( ( P @ W13 @ W24 )
=> ( P @ W13 @ ( cons @ B @ E2 @ W24 ) ) )
=> ( P @ W1 @ W22 ) ) ) ) ).
% list_2pre_induct
thf(fact_4978_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( L
!= ( cons @ A @ X3 @ Xs4 ) ) ) ).
% neq_NilE
thf(fact_4979_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N22: A,R22: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N22 @ R22 ) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_4980_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F: A > B > C,A6: A,As2: list @ A,B2: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As2 ) @ ( cons @ B @ B2 @ 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_4981_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
=> ( ! [V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X12: A,L1: list @ A,X23: A,L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_4982_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,As2: list @ A,B2: 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 @ As2 ) @ ( cons @ B @ B2 @ 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_4983_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_4984_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,Sl: list @ A,X3: A,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs4 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_4985_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,X3: A,Xs4: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
=> ( ! [R8: A > A > $o,Xs4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs4 @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,V3: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_4986_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_4987_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ! [X3: A,Xs4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( ( ( X @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs4 @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) )
& ( ~ ( X @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs4 ) @ Ys2 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V3 @ Va ) )
=> ( Y
!= ( cons @ A @ V3 @ Va ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_4988_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R4: A > A > $o,V: A,Va2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R4 @ ( nil @ A ) @ ( cons @ A @ V @ Va2 ) )
= ( cons @ A @ V @ Va2 ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_4989_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_4990_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E2: A,L3: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E2 @ L3 ) ) )
=> ( P @ L ) ) ) ).
% length_compl_induct
thf(fact_4991_list__decomp__1,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( one_one @ nat ) )
=> ? [A6: A] :
( L
= ( cons @ A @ A6 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_4992_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_4993_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_4994_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A6: A,B2: A] :
( L
= ( cons @ A @ A6 @ ( cons @ A @ B2 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_4995_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_4996_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 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( ( ( X @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs4 @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) )
& ( ~ ( X @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs4 ) @ Ys2 ) ) ) ) )
=> ~ ( 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 @ Y2 @ Ys2 ) ) ) ) ) ) )
=> ( ( ( 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_4997_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_4998_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ 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_4999_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_5000_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ 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_5001_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R2 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_5002_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_5003_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N2: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N2 @ Ns ) ) @ ( lenlex @ A @ R2 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N2 ) @ R2 ) )
| ( ( M = N2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_5004_listrel__Nil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_5005_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_5006_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_5007_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [Y2: B,Ys2: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys2 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_5008_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_5009_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y2: B,Xs4: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Ys2: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys2 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_5010_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X2: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_5011_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N4 ) @ ( nth @ B @ Ys @ N4 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_5012_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( ( 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 @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_5013_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_5014_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I4: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I4 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_5015_set__mergesort__by__rel,axiom,
! [A: $tType,R4: A > A > $o,Xs: list @ A] :
( ( set2 @ A @ ( mergesort_by_rel @ A @ R4 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mergesort_by_rel
thf(fact_5016_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_5017_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_5018_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_5019_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R4: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R4 @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_5020_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5021_nth__image__indices,axiom,
! [A: $tType,L: list @ A] :
( ( image2 @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( set2 @ A @ L ) ) ).
% nth_image_indices
thf(fact_5022_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( insert2 @ B @ X @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_5023_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_5024_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ L ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I4 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_5025_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_5026_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F2: B > A,X: B,L: list @ B] :
( ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F2 @ ( nth @ B @ L @ I4 ) )
= ( F2 @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_5027_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F2: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L4 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F2 @ ( nth @ A @ L @ I3 ) )
= ( F2 @ ( nth @ A @ L4 @ I3 ) ) ) )
=> ( ( image2 @ A @ B @ F2 @ ( set2 @ A @ L ) )
= ( image2 @ A @ B @ F2 @ ( set2 @ A @ L4 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_5028_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_5029_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_5030_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_5031_is__empty__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( is_empty @ A @ ( set2 @ A @ Xs ) )
= ( null @ A @ Xs ) ) ).
% is_empty_set
thf(fact_5032_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_5033_set__map__filter,axiom,
! [B: $tType,A: $tType,G: B > ( option @ A ),Xs: list @ B] :
( ( set2 @ A @ ( map_filter @ B @ A @ G @ Xs ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ Xs ) )
& ( ( G @ X2 )
= ( some @ A @ Y3 ) ) ) ) ) ).
% set_map_filter
thf(fact_5034_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_5035_Un__set__drop__extend,axiom,
! [A: $tType,J2: nat,L: list @ ( set @ A )] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
=> ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J2 @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% Un_set_drop_extend
thf(fact_5036_nth__zip,axiom,
! [A: $tType,B: $tType,I2: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys @ I2 ) ) ) ) ) ).
% nth_zip
thf(fact_5037_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,A7: list @ A,B4: list @ B] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A7 )
= ( size_size @ ( list @ B ) @ B4 ) )
=> ( ( ( zip @ A @ B @ A4 @ B3 )
= ( zip @ A @ B @ A7 @ B4 ) )
= ( ( A4 = A7 )
& ( B3 = B4 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_5038_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_5039_zip__inj,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,A7: list @ A,B4: list @ B] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A7 )
= ( size_size @ ( list @ B ) @ B4 ) )
=> ( ( ( zip @ A @ B @ A4 @ B3 )
= ( zip @ A @ B @ A7 @ B4 ) )
=> ( ( A4 = A7 )
& ( B3 = B4 ) ) ) ) ) ).
% zip_inj
thf(fact_5040_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X3: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs5 ) )
=> ! [Y2: B,Ys4: list @ B] :
( ( Ys
= ( cons @ B @ Y2 @ Ys4 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_5041_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_5042_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_5043_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_5044_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_5045_drop__eq__ConsD,axiom,
! [A: $tType,N2: nat,Xs: list @ A,X: A,Xs6: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( cons @ A @ X @ Xs6 ) )
=> ( ( drop @ A @ ( suc @ N2 ) @ Xs )
= Xs6 ) ) ).
% drop_eq_ConsD
thf(fact_5046_set__zip__cart,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L4: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L4 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu: A] : ( set2 @ B @ L4 ) ) ) ) ).
% set_zip_cart
thf(fact_5047_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list @ ( product_prod @ A @ B )] :
~ ! [L1: list @ A,L22: list @ B] :
( ( L
= ( zip @ A @ B @ L1 @ L22 ) )
=> ( ( ( size_size @ ( list @ A ) @ L1 )
= ( size_size @ ( list @ B ) @ L22 ) )
=> ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ L )
!= ( size_size @ ( list @ B ) @ L22 ) ) ) ) ).
% pair_list_split
thf(fact_5048_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_5049_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_5050_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_5051_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_5052_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N2: nat,L: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N2 @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ N2 @ I4 )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
& ( X
= ( nth @ A @ L @ I4 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_5053_in__set__zip,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N4: nat] :
( ( ( nth @ A @ Xs @ N4 )
= ( product_fst @ A @ B @ P3 ) )
& ( ( nth @ B @ Ys @ N4 )
= ( product_snd @ A @ B @ P3 ) )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_5054_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 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs4 @ ( cons @ A @ Y2 @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs4 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_5055_set__drop__conv,axiom,
! [A: $tType,N2: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N2 @ L ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I4: nat] :
( ( Uu
= ( nth @ A @ L @ I4 ) )
& ( ord_less_eq @ nat @ N2 @ I4 )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_5056_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] :
? [I4: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_5057_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( 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
@ ^ [Y3: A,Z3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R2 )
@ X2 ) ) ) ) ).
% listrel_iff_zip
thf(fact_5058_drop__last__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( cons @ A @ ( last @ A @ L ) @ ( nil @ A ) ) ) ) ).
% drop_last_conv
thf(fact_5059_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [I4: nat] :
( ( Uu
= ( F4 @ ( nth @ B @ L2 @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_5060_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_5061_Misc_Olast__in__set,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% Misc.last_in_set
thf(fact_5062_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A4: B] :
( ( list_collect_set @ B @ A @ F2 @ ( cons @ B @ A4 @ ( nil @ B ) ) )
= ( F2 @ A4 ) ) ).
% list_collect_set_simps(2)
thf(fact_5063_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_5064_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A4: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F2 @ ( cons @ B @ A4 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ A4 ) @ ( list_collect_set @ B @ A @ F2 @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_5065_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_5066_lists__of__len__fin2,axiom,
! [A: $tType,P: set @ A,N2: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L2: list @ A] :
( N2
= ( size_size @ ( list @ A ) @ L2 ) ) ) ) ) ) ).
% lists_of_len_fin2
thf(fact_5067_lists__of__len__fin1,axiom,
! [A: $tType,P: set @ A,N2: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L2: list @ A] :
( ( size_size @ ( list @ A ) @ L2 )
= N2 ) ) ) ) ) ).
% lists_of_len_fin1
thf(fact_5068_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [A8: B] :
( ( Uu
= ( F4 @ A8 ) )
& ( member @ B @ A8 @ ( set2 @ B @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_5069_last__take__nth__conv,axiom,
! [A: $tType,N2: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N2 @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_5070_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_5071_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ 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_5072_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
= ( fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ Xs @ A3 ) ) ).
% union_set_fold
thf(fact_5073_slice__def,axiom,
! [A: $tType] :
( ( slice @ A )
= ( ^ [From2: nat,To2: nat,List: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List ) ) ) ) ).
% slice_def
thf(fact_5074_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_5075_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_5076_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_5077_Union__take__drop__id,axiom,
! [A: $tType,N2: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N2 @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N2 @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_5078_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [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 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_5079_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I4 @ X )
= ( take @ A @ I4 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I4 ) @ ( nth @ A @ Y @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5080_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,J2: nat,X: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J2 @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J2 @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J2 @ Ys ) @ ( cons @ B @ Y @ ( drop @ B @ J2 @ 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_5081_take__butlast__conv,axiom,
! [A: $tType,L: list @ A] :
( ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ L ) ) ).
% take_butlast_conv
thf(fact_5082_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X2: list @ A] : X2 ) ) ).
% empty_append_eq_id
thf(fact_5083_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L12: list @ A,E12: A,E23: A,L23: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_5084_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list @ A,E12: A,E23: A,L23: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_5085_list__se__match_I1_J,axiom,
! [A: $tType,L12: list @ A,L23: list @ A,A4: A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L12
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_5086_list__se__match_I2_J,axiom,
! [A: $tType,L23: list @ A,L12: list @ A,A4: A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_5087_list__se__match_I3_J,axiom,
! [A: $tType,L12: list @ A,A4: A,L23: list @ A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_5088_list__se__match_I4_J,axiom,
! [A: $tType,L23: list @ A,A4: A,L12: list @ A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_5089_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E4: A,L12: list @ A,E5: A,L23: list @ A] :
( ( ( cons @ A @ E4 @ ( nil @ A ) )
= ( append @ A @ L12 @ ( cons @ A @ E5 @ L23 ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E5 = E4 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_5090_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list @ A,E5: A,L23: list @ A,E4: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E5 @ L23 ) )
= ( cons @ A @ E4 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E5 = E4 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_5091_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),L: list @ B,L4: list @ B] :
( ( list_collect_set @ B @ A @ F2 @ ( append @ B @ L @ L4 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F2 @ L ) @ ( list_collect_set @ B @ A @ F2 @ L4 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_5092_op__conc__empty__img__id,axiom,
! [A: $tType,L5: set @ ( list @ A )] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L5 )
= L5 ) ).
% op_conc_empty_img_id
thf(fact_5093_nth__append__first,axiom,
! [A: $tType,I2: nat,L: list @ A,L4: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L4 ) @ I2 )
= ( nth @ A @ L @ I2 ) ) ) ).
% nth_append_first
thf(fact_5094_lexord__cons__cons,axiom,
! [A: $tType,A4: A,X: list @ A,B3: A,Y: list @ A,R2: 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 @ R2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
| ( ( A4 = B3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_5095_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,X: A] :
( ( Ys
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_5096_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_5097_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
=> ~ ! [Xl: A] :
( L
!= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_5098_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
= ( ? [Xl2: A] :
( L
= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_5099_list__match__lel__lel,axiom,
! [A: $tType,C12: list @ A,Qs: A,C23: list @ A,C13: list @ A,Qs2: A,C24: list @ A] :
( ( ( append @ A @ C12 @ ( cons @ A @ Qs @ C23 ) )
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C24 ) ) )
=> ( ! [C21: list @ A] :
( ( C12
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C24
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C23 ) ) ) )
=> ( ( ( C13 = C12 )
=> ( ( Qs2 = Qs )
=> ( C24 != C23 ) ) )
=> ~ ! [C212: list @ A] :
( ( C13
= ( append @ A @ C12 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C23
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C24 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_5100_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X2: list @ A,Y3: list @ A] :
? [A8: A,V4: list @ A] :
( ( Y3
= ( append @ A @ X2 @ ( cons @ A @ A8 @ V4 ) ) )
| ? [U3: list @ A,B6: A,C5: A,W3: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R5 )
& ( X2
= ( append @ A @ U3 @ ( cons @ A @ B6 @ W3 ) ) )
& ( Y3
= ( append @ A @ U3 @ ( cons @ A @ C5 @ Z3 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_5101_lexord__append__left__rightI,axiom,
! [A: $tType,A4: A,B3: A,R2: 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 ) @ R2 )
=> ( 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 @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_5102_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_5103_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs2 )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs2
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs2 )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_5104_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs2 ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs2
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs2 )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_5105_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X3: A,Y2: B] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) )
=> ( ! [X3: A,Xs4: list @ A,Y2: B] :
( ( Xs4
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Y2: B,Ys2: list @ B] :
( ( Ys2
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( ! [X3: A,Xs4: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs4 @ Ys2 )
=> ( ( Xs4
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_5106_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X2: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_5107_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs4: list @ A] : ( P @ ( append @ A @ Xs4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Xs4: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs4 @ Ys2 )
=> ( P @ ( append @ A @ Xs4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_5108_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll2: list @ A,E2: A] :
( L
!= ( append @ A @ Ll2 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_5109_in__set__list__format,axiom,
! [A: $tType,E4: A,L: list @ A] :
( ( member @ A @ E4 @ ( set2 @ A @ L ) )
=> ~ ! [L1: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ E4 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_5110_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X = Y )
=> ! [L1: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_5111_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R1 )
= ( append @ A @ S22 @ R22 ) )
=> ? [R1p: list @ A] :
( R22
= ( append @ A @ R1p @ R1 ) ) ) ) ).
% list_rest_coinc
thf(fact_5112_set__union__code,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( set2 @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% set_union_code
thf(fact_5113_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_5114_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A6: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B2 ) @ R2 )
| ( A6 = B2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A6 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_5115_drop__take__drop__unsplit,axiom,
! [A: $tType,I2: nat,J2: nat,L: list @ A] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( append @ A @ ( drop @ A @ I2 @ ( take @ A @ J2 @ L ) ) @ ( drop @ A @ J2 @ L ) )
= ( drop @ A @ I2 @ L ) ) ) ).
% drop_take_drop_unsplit
thf(fact_5116_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_5117_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_5118_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A3 ) ) ) ).
% butlast_subset
thf(fact_5119_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs2: list @ A] :
( ! [X3: A,Y2: A,Z4: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_5120_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Ys3: list @ A,Y3: A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_5121_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L3: list @ A,E2: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L3 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
=> ( P @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_5122_slice__prepend,axiom,
! [A: $tType,I2: nat,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I2 @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I2 @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_5123_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ Y ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_5124_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_5125_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X2: A,Y3: A,Xs7: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs7 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_5126_take__minus__one__conv__butlast,axiom,
! [A: $tType,N2: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( take @ A @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ ( take @ A @ N2 @ L ) ) ) ) ).
% take_minus_one_conv_butlast
thf(fact_5127_butlast__upd__last__eq,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( list_update @ A @ ( butlast @ A @ L ) @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ X )
= ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% butlast_upd_last_eq
thf(fact_5128_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N4: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N4 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N4 )
& ? [Xys2: list @ A,X2: A,Y3: A,Xs7: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs7 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_5129_take__update__last,axiom,
! [A: $tType,N2: nat,List2: list @ A,X: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ List2 ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N2 ) @ List2 ) @ N2 @ X )
= ( append @ A @ ( take @ A @ N2 @ List2 ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_5130_take__update,axiom,
! [A: $tType,N2: nat,L: list @ A,I2: nat,X: A] :
( ( take @ A @ N2 @ ( list_update @ A @ L @ I2 @ X ) )
= ( list_update @ A @ ( take @ A @ N2 @ L ) @ I2 @ X ) ) ).
% take_update
thf(fact_5131_drop__upd__irrelevant,axiom,
! [A: $tType,M: nat,N2: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( drop @ A @ N2 @ ( list_update @ A @ L @ M @ X ) )
= ( drop @ A @ N2 @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_5132_nth__update__invalid,axiom,
! [A: $tType,I2: nat,L: list @ A,J2: nat,X: A] :
( ~ ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J2 @ X ) @ I2 )
= ( nth @ A @ L @ I2 ) ) ) ).
% nth_update_invalid
thf(fact_5133_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I2: nat,X: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I2 @ X )
= ( butlast @ A @ ( list_update @ A @ L @ I2 @ X ) ) ) ).
% butlast_update'
thf(fact_5134_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I2: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I2 @ X ) @ ( list_update @ B @ Ys @ I2 @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I2 @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_5135_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I2: nat,Xy2: product_prod @ A @ B] :
( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I2 @ Xy2 )
= ( zip @ A @ B @ ( list_update @ A @ Xs @ I2 @ ( product_fst @ A @ B @ Xy2 ) ) @ ( list_update @ B @ Ys @ I2 @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% update_zip
thf(fact_5136_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_5137_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_5138_in__set__upd__eq__aux,axiom,
! [A: $tType,I2: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y ) ) )
= ( ( X = Y )
| ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y3 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_5139_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list @ A,I2: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y ) ) )
=> ( ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( X != Y ) )
=> ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_5140_in__set__upd__eq,axiom,
! [A: $tType,I2: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y ) ) )
= ( ( X = Y )
| ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y3 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_5141_nth__list__update_H,axiom,
! [A: $tType,I2: nat,J2: nat,L: list @ A,X: A] :
( ( ( ( I2 = J2 )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I2 @ X ) @ J2 )
= X ) )
& ( ~ ( ( I2 = J2 )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I2 @ X ) @ J2 )
= ( nth @ A @ L @ J2 ) ) ) ) ).
% nth_list_update'
thf(fact_5142_insert__swap__set__eq,axiom,
! [A: $tType,I2: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert2 @ A @ ( nth @ A @ L @ I2 ) @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ X ) ) )
= ( insert2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_5143_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N2: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ N2 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_5144_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us: list @ A,Z3: A,Z11: A,Vs: list @ A] :
( ( Xs3
= ( append @ A @ Us @ ( cons @ A @ Z3 @ Vs ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Z11 ) @ R5 )
& ( Ys3
= ( append @ A @ Us @ ( cons @ A @ Z11 @ Vs ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_5145_foldl__list__update,axiom,
! [B: $tType,A: $tType,N2: nat,Xs: list @ A,F2: B > A > B,A4: B,X: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F2 @ A4 @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( foldl @ B @ A @ F2 @ ( F2 @ ( foldl @ B @ A @ F2 @ A4 @ ( take @ A @ N2 @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N2 ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_5146_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I4: nat,X2: A] : ( suc @ I4 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_5147_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_5148_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_5149_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_5150_foldl__A1__eq,axiom,
! [A: $tType,F2: A > A > A,N2: A,I2: A,Ww: list @ A] :
( ! [E2: A] :
( ( F2 @ N2 @ E2 )
= E2 )
=> ( ! [E2: A] :
( ( F2 @ E2 @ N2 )
= E2 )
=> ( ! [A6: A,B2: A,C3: A] :
( ( F2 @ A6 @ ( F2 @ B2 @ C3 ) )
= ( F2 @ ( F2 @ A6 @ B2 ) @ C3 ) )
=> ( ( foldl @ A @ A @ F2 @ I2 @ Ww )
= ( F2 @ I2 @ ( foldl @ A @ A @ F2 @ N2 @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_5151_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F2: A > B > B,A4: A,B3: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( ( F2 @ A4
@ ( foldl @ B @ A
@ ^ [A8: B,B6: A] : ( F2 @ B6 @ A8 )
@ B3
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A8: B,B6: A] : ( F2 @ B6 @ A8 )
@ ( F2 @ A4 @ B3 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_5152_distinct__match,axiom,
! [A: $tType,Al: list @ A,E4: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E4 @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E4 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E4 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_5153_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > C > A,G: 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 ) )
@ ^ [A8: A,B6: B,X2: C] : ( product_Pair @ A @ B @ ( F2 @ A8 @ X2 ) @ ( G @ A8 @ B6 @ X2 ) ) )
@ ( product_Pair @ A @ B @ A4 @ B3 )
@ Xs ) )
= ( foldl @ A @ C @ F2 @ A4 @ Xs ) ) ).
% fst_foldl
thf(fact_5154_foldl__conc__empty__eq,axiom,
! [A: $tType,I2: list @ A,Ww: list @ ( list @ A )] :
( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I2 @ Ww )
= ( append @ A @ I2 @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% foldl_conc_empty_eq
thf(fact_5155_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_5156_foldl__un__empty__eq,axiom,
! [A: $tType,I2: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I2 @ Ww )
= ( sup_sup @ ( set @ A ) @ I2 @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_5157_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list @ A,Xs: list @ B,F2: C > A > C,B3: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B6: C] :
( product_case_prod @ B @ A @ C
@ ^ [X2: B] : ( F2 @ B6 ) )
@ B3
@ ( zip @ B @ A @ Xs @ Ys ) )
= ( foldl @ C @ A @ F2 @ B3 @ Ys ) ) ) ).
% foldl_snd_zip
thf(fact_5158_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list @ A,I: ( set @ A ) > B > $o,Sigma_0: B,F2: B > A > B] :
( ( distinct @ A @ S )
=> ( ( I @ ( 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 ) )
=> ( ( I @ It @ Sigma )
=> ( I @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F2 @ Sigma @ X3 ) ) ) ) )
=> ( I @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F2 @ Sigma_0 @ S ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_5159_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( set2 @ A @ Ys3 )
= X )
& ( distinct @ A @ Ys3 ) ) ) ) ).
% distinct_finite_set
thf(fact_5160_distinct__length__le,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% distinct_length_le
thf(fact_5161_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list @ A,I2: nat] :
( ( distinct @ A @ Pq )
=> ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I2 @ ( last @ A @ Pq ) ) ) ) ) ).
% distinct_butlast_swap
thf(fact_5162_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [X3: A] :
( ( Xs
= ( cons @ A @ X3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) )
=> ~ ! [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 @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_5163_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 ) )
=> ~ ! [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 @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_5164_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_5165_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F2: Sigma2 > A > Sigma2] :
( ( I @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I @ Sigma @ L1 @ ( cons @ A @ X3 @ L22 ) )
=> ( I @ ( F2 @ Sigma @ X3 ) @ ( append @ A @ L1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( I @ ( foldl @ Sigma2 @ A @ F2 @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_5166_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F2: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I @ Sigma @ L1 @ ( cons @ A @ X3 @ L22 ) )
=> ( I @ ( F2 @ Sigma @ X3 ) @ ( append @ A @ L1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I @ Sigma @ L0 @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F2 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_5167_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F2: Sigma2 > A > Sigma2] :
( ( I @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I @ Sigma @ ( cons @ A @ X3 @ L22 ) )
=> ( I @ ( F2 @ Sigma @ X3 ) @ L22 ) ) )
=> ( I @ ( foldl @ Sigma2 @ A @ F2 @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_5168_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F2: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I @ Sigma @ ( cons @ A @ X3 @ L22 ) )
=> ( I @ ( F2 @ Sigma @ X3 ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I @ Sigma @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F2 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_5169_finite__set__image,axiom,
! [A: $tType,A3: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) )
=> ( ! [Xs4: list @ A] :
( ( member @ ( list @ A ) @ Xs4 @ A3 )
=> ( distinct @ A @ Xs4 ) )
=> ( finite_finite2 @ ( list @ A ) @ A3 ) ) ) ).
% finite_set_image
thf(fact_5170_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y2: A,Ys2: list @ A] :
( ( distinct @ A @ Ys2 )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Ys2 ) )
=> ! [Zs: list @ A] :
( Xs
!= ( append @ A @ Ys2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_5171_foldl__length__aux,axiom,
! [A: $tType,A4: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I4: nat,X2: A] : ( suc @ I4 )
@ A4
@ L )
= ( plus_plus @ nat @ A4 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_5172_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_5173_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ~ ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ! [Us2: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us2 @ ( cons @ A @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_5174_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Us3: list @ A,Vs3: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
=> ( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ Y @ Vs3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_5175_distinct__finite__subset,axiom,
! [A: $tType,X: set @ A] :
( ( finite_finite2 @ A @ X )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ X )
& ( distinct @ A @ Ys3 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_5176_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_5177_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_5178_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_5179_distinct__sorted__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I2: nat,J2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( distinct @ A @ L )
=> ( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ord_less @ A @ ( nth @ A @ L @ I2 ) @ ( nth @ A @ L @ J2 ) ) ) ) ) ) ) ).
% distinct_sorted_mono
thf(fact_5180_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I2: nat,J2: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ A @ ( nth @ A @ L @ I2 ) @ ( nth @ A @ L @ J2 ) )
= ( ord_less @ nat @ I2 @ J2 ) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
thf(fact_5181_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I2: nat,J2: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I2 ) @ ( nth @ A @ L @ J2 ) )
= ( ord_less_eq @ nat @ I2 @ J2 ) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
thf(fact_5182_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A4: A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A4 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ I2 ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I2 @ A4 ) ) ) ) ).
% distinct_list_update
thf(fact_5183_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs3: list @ A,I2: nat,J2: nat] :
( ( distinct @ A @ Vs3 )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I2 @ Vs3 ) ) @ ( set2 @ A @ ( drop @ A @ J2 @ Vs3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_5184_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y3: A,N4: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N4 ) @ Y3 ) @ R2 )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N4 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_5185_mergesort__remdups__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_remdups @ A @ L ) )
& ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
= ( set2 @ A @ L ) ) ) ) ).
% mergesort_remdups_correct
thf(fact_5186_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 ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y2 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% inv_image_partition
thf(fact_5187_merge__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L12: list @ A,L23: list @ A] :
( ( ( distinct @ A @ L12 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L12 ) )
=> ( ( ( distinct @ A @ L23 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L23 ) )
=> ( ( distinct @ A @ ( merge @ A @ L12 @ L23 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge @ A @ L12 @ L23 ) )
& ( ( set2 @ A @ ( merge @ A @ L12 @ L23 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L12 ) @ ( set2 @ A @ L23 ) ) ) ) ) ) ) ).
% merge_correct
thf(fact_5188_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L23: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L23 )
= L23 ) ) ).
% merge.simps(1)
thf(fact_5189_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X1: A,X22: A,L12: list @ A,L23: list @ A] :
( ( ( ord_less @ A @ X1 @ X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X22 @ L23 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ ( cons @ A @ X22 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X22 )
=> ( ( ( X1 = X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X22 @ L23 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X1 != X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X22 @ L23 ) )
= ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_5190_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va2: list @ A] :
( ( merge @ A @ ( cons @ A @ V @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V @ Va2 ) ) ) ).
% merge.simps(2)
thf(fact_5191_merge_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V3: A,Va: list @ A] :
( ( X
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V3 @ Va ) ) ) )
=> ~ ! [X12: A,L1: list @ A] :
( ( X
= ( cons @ A @ X12 @ L1 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_5192_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R5: 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,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R5 ) ) ) ) ) ) ).
% listrel_def
thf(fact_5193_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 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs4 @ ( cons @ A @ Y2 @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs4 ) @ Ys2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs4 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_5194_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: list @ A,Y3: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X2 @ Y3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_5195_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,Acc2: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A ),L3: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc2: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_5196_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_5197_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_5198_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,L1: list @ A] :
( ( X
= ( cons @ A @ X12 @ L1 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ( ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X12 @ L1 ) @ L22 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_5199_zipf__zip,axiom,
! [A: $tType,B: $tType,L12: list @ A,L23: list @ B] :
( ( ( size_size @ ( list @ A ) @ L12 )
= ( size_size @ ( list @ B ) @ L23 ) )
=> ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L12 @ L23 )
= ( zip @ A @ B @ L12 @ L23 ) ) ) ).
% zipf_zip
thf(fact_5200_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_5201_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,A4: A,As: list @ A,B3: B,Bs: list @ B] :
( ( zipf @ A @ B @ C @ F2 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) )
= ( cons @ C @ ( F2 @ A4 @ B3 ) @ ( zipf @ A @ B @ C @ F2 @ As @ Bs ) ) ) ).
% zipf.simps(2)
thf(fact_5202_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B > C] :
( ( zipf @ A @ B @ C @ F2 @ ( nil @ A ) @ ( nil @ B ) )
= ( nil @ C ) ) ).
% zipf.simps(1)
thf(fact_5203_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( nil @ C ) ) ) )
=> ( ! [A6: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ Bs2 ) )
=> ( Y
!= ( cons @ C @ ( X @ A6 @ B2 ) @ ( zipf @ A @ B @ C @ X @ As2 @ Bs2 ) ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V3: B,Va: list @ B] :
( Xb
= ( cons @ B @ V3 @ Va ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% zipf.elims
thf(fact_5204_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,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X @ A6 @ B2 ) @ ( zipf @ A @ B @ C @ X @ As2 @ 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 @ As2 ) @ ( cons @ B @ B2 @ 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_5205_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,Acc2: 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 @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: 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 @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc2: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc2 ) @ Ls )
=> ( P @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_5206_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I2: nat,L: list @ A,F2: A > B,X: B] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F2 @ L ) @ I2 @ X )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F2 @ ( nth @ A @ L @ I2 ) @ X ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_5207_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_5208_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_5209_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_5210_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs2: list @ ( product_prod @ A @ B )] :
( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 ) )
= Zs2 ) ).
% zip_map_fst_snd
thf(fact_5211_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% sorted_wrt_map_linord
thf(fact_5212_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,A4: C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( cons @ C @ A4 @ ( nil @ C ) ) ) )
= ( F2 @ ( X @ A4 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_5213_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_5214_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,A4: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( cons @ C @ A4 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ ( X @ A4 ) ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_5215_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,L: list @ C,L4: list @ C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( append @ C @ L @ L4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L4 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_5216_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A3: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A3 )
=> ( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F2 @ A3 ) @ ( map @ A @ B @ F2 @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_5217_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_5218_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,Y3: C] : ( product_Pair @ A @ B @ X2 @ ( F2 @ Y3 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_5219_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X2: D,Y3: C] : ( F2 @ ( product_Pair @ B @ C @ ( G @ X2 ) @ Y3 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_5220_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X2: B,Y3: D] : ( F2 @ ( product_Pair @ B @ C @ X2 @ ( G @ Y3 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_5221_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G: 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,Y3: D] : ( product_Pair @ A @ B @ ( F2 @ X2 ) @ ( G @ Y3 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ ( map @ D @ B @ G @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_5222_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F2 ) )
= ( ^ [L2: list @ A] : ( image2 @ A @ B @ F2 @ ( set2 @ A @ L2 ) ) ) ) ).
% set_oo_map_alt
thf(fact_5223_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,L: list @ B,L4: list @ B,I2: nat] :
( ( ( map @ B @ A @ F2 @ L )
= ( map @ B @ A @ F2 @ L4 ) )
=> ( ( F2 @ ( nth @ B @ L @ I2 ) )
= ( F2 @ ( nth @ B @ L4 @ I2 ) ) ) ) ).
% map_eq_nth_eq
thf(fact_5224_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Ls2: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls2 )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L2: list @ B,L6: list @ B] :
( ( Ls2
= ( append @ B @ L2 @ L6 ) )
& ( ( map @ B @ A @ F2 @ L2 )
= Fl )
& ( ( map @ B @ A @ F2 @ L6 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_5225_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F2: B > A,Ls2: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F2 @ Ls2 ) )
= ( ? [L2: list @ B,L6: list @ B] :
( ( Ls2
= ( append @ B @ L2 @ L6 ) )
& ( ( map @ B @ A @ F2 @ L2 )
= Fl )
& ( ( map @ B @ A @ F2 @ L6 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_5226_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F2: B > A,Ls2: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls2 )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L3: list @ B,L7: list @ B] :
( ( Ls2
= ( append @ B @ L3 @ L7 ) )
=> ( ( ( map @ B @ A @ F2 @ L3 )
= Fl )
=> ( ( map @ B @ A @ F2 @ L7 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_5227_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F2: B > A,Ls2: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F2 @ Ls2 ) )
=> ~ ! [L3: list @ B,L7: list @ B] :
( ( Ls2
= ( append @ B @ L3 @ L7 ) )
=> ( ( ( map @ B @ A @ F2 @ L3 )
= Fl )
=> ( ( map @ B @ A @ F2 @ L7 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_5228_distinct__mapI,axiom,
! [A: $tType,B: $tType,F2: B > A,L: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L ) )
=> ( distinct @ B @ L ) ) ).
% distinct_mapI
thf(fact_5229_map__eq__consE,axiom,
! [B: $tType,A: $tType,F2: B > A,Ls2: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls2 )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A6: B,L3: list @ B] :
( ( Ls2
= ( cons @ B @ A6 @ L3 ) )
=> ( ( ( F2 @ A6 )
= Fa )
=> ( ( map @ B @ A @ F2 @ L3 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_5230_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list @ A,F2: B > A,Ww: list @ B,A4: B] :
( ( W
= ( map @ B @ A @ F2 @ Ww ) )
=> ( ( cons @ A @ ( F2 @ A4 ) @ W )
= ( map @ B @ A @ F2 @ ( cons @ B @ A4 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_5231_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list @ A,L: list @ A,F2: B > A,Ww: list @ B,A4: B] :
( ( ( append @ A @ W @ L )
= ( append @ A @ ( map @ B @ A @ F2 @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F2 @ A4 ) @ ( append @ A @ W @ L ) )
= ( append @ A @ ( map @ B @ A @ F2 @ ( cons @ B @ A4 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_5232_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,L: list @ B,X: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L ) )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member @ B @ X @ ( set2 @ B @ L ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L ) )
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_5233_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_5234_lists__image__witness,axiom,
! [A: $tType,B: $tType,X: list @ A,F2: B > A,Q2: set @ B] :
( ( member @ ( list @ A ) @ X @ ( lists @ A @ ( image2 @ B @ A @ F2 @ Q2 ) ) )
=> ~ ! [Xo2: list @ B] :
( ( member @ ( list @ B ) @ Xo2 @ ( lists @ B @ Q2 ) )
=> ( X
!= ( map @ B @ A @ F2 @ Xo2 ) ) ) ) ).
% lists_image_witness
thf(fact_5235_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_5236_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_5237_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,Y3: B,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C @ Y3 @ Z3 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs2 ) ) ) ).
% zip_assoc
thf(fact_5238_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 ) )
@ ^ [Y3: 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 @ Y3 @ Z3 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs2 ) ) ) ) ).
% zip_left_commute
thf(fact_5239_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X2: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X2 ) )
@ ( zip @ B @ A @ Ys3 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_5240_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L2: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F4 @ L2 ) ) ) ) ) ).
% list_collect_set_as_map
thf(fact_5241_distinct__idx,axiom,
! [B: $tType,A: $tType,F2: B > A,L: list @ B,I2: nat,J2: nat] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F2 @ ( nth @ B @ L @ I2 ) )
= ( F2 @ ( nth @ B @ L @ J2 ) ) )
=> ( I2 = J2 ) ) ) ) ) ).
% distinct_idx
thf(fact_5242_map__upd__eq,axiom,
! [B: $tType,A: $tType,I2: nat,L: list @ A,F2: A > B,X: A] :
( ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F2 @ ( nth @ A @ L @ I2 ) )
= ( F2 @ X ) ) )
=> ( ( map @ A @ B @ F2 @ ( list_update @ A @ L @ I2 @ X ) )
= ( map @ A @ B @ F2 @ L ) ) ) ).
% map_upd_eq
thf(fact_5243_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs2 )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_5244_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_5245_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_5246_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F2 @ X ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_5247_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_5248_map__by__foldl,axiom,
! [B: $tType,A: $tType,F2: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L2: list @ B,X2: A] : ( append @ B @ L2 @ ( cons @ B @ ( F2 @ X2 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F2 @ L ) ) ).
% map_by_foldl
thf(fact_5249_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ B,Ys: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ).
% map_snd_zip_take
thf(fact_5250_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,Acc2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ~ ( 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 @ Acc2 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [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 @ Acc2 ) ) ) )
=> ~ ( 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 @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X ) @ Ls ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_5251_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) ) )
=> ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls2 ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_5252_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_5253_merge__list_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ) ).
% merge_list.simps(1)
thf(fact_5254_merge__list_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= L ) ) ).
% merge_list.simps(2)
thf(fact_5255_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A )] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ).
% merge_list.simps(3)
thf(fact_5256_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ).
% merge_list.simps(4)
thf(fact_5257_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls2 ) ) ) ).
% merge_list.simps(5)
thf(fact_5258_merge__list_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L3 ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [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 @ Acc2 ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X ) @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_5259_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs3: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X2: A] : ( cons @ A @ X2 @ ( nil @ A ) )
@ Xs3 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_5260_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_5261_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ) ).
% merge_list.psimps(4)
thf(fact_5262_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ).
% merge_list.psimps(3)
thf(fact_5263_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_5264_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,Y8: 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 @ Y8 ) @ Ys ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_5265_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_5266_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_5267_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_5268_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_5269_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_5270_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_5271_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_5272_ran__map__of,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( map_of @ B @ A @ Xs ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% ran_map_of
thf(fact_5273_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P3 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P3 ) @ ( some @ B @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% map_of.simps(2)
thf(fact_5274_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 ) )
=> ? [Ys2: list @ ( product_prod @ B @ A ),Zs: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys2 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs ) ) )
& ( ( map_of @ B @ A @ Ys2 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_5275_map__to__set__map__of,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( map_to_set @ A @ B @ ( map_of @ A @ B @ L ) )
= ( set2 @ ( product_prod @ A @ B ) @ L ) ) ) ).
% map_to_set_map_of
thf(fact_5276_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M: A > ( option @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( ( map_of @ A @ B @ L )
= M )
= ( ( set2 @ ( product_prod @ A @ B ) @ L )
= ( map_to_set @ A @ B @ M ) ) ) ) ).
% map_of_map_to_set
thf(fact_5277_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 )
@ ^ [K5: A] : ( product_Pair @ A @ B @ K5 @ ( F2 @ K5 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F2 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_5278_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K5: A] : ( product_Pair @ A @ B @ K5 @ ( the2 @ B @ ( M @ K5 ) ) )
@ Xs ) )
= M ) ) ).
% map_of_map_keys
thf(fact_5279_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,T5: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T5 @ 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 )
@ ^ [K5: A] : ( product_Pair @ B @ C @ ( F2 @ K5 ) ) )
@ T5 )
@ ( F2 @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_5280_Misc_Oran__distinct,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% Misc.ran_distinct
thf(fact_5281_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_5282_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_5283_image__mset__map__of,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( image_mset @ A @ B
@ ^ [I4: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I4 ) )
@ ( mset @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
= ( mset @ B @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) ) ) ) ).
% image_mset_map_of
thf(fact_5284_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y3 ) @ ( product_fst @ A @ B @ X2 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% sorted_wrt_map_rev_linord
thf(fact_5285_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_5286_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_5287_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_5288_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M ) )
=> ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M ) )
= ( map_of @ A @ B @ M ) ) ) ).
% map_of_rev_distinct
thf(fact_5289_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F2: A > B > A,A4: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F2 ) @ A4 @ Xs )
= ( foldl @ A @ B @ F2 @ A4 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_5290_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: 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 ) @ Ys3 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_5291_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys2: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys2 ) )
=> ( ! [Ys2: list @ A,Zs: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys2 != Zs )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Zs ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_5292_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_5293_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls2: list @ ( list @ A ),As: list @ ( list @ A )] :
( ! [L3: list @ A] :
( ( member @ ( list @ A ) @ L3 @ ( set2 @ ( list @ A ) @ Ls2 ) )
=> ( ( distinct @ A @ L3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L3 ) ) )
=> ( ! [L3: list @ A] :
( ( member @ ( list @ A ) @ L3 @ ( set2 @ ( list @ A ) @ As ) )
=> ( ( distinct @ A @ L3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L3 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As @ Ls2 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As @ Ls2 ) )
& ( ( set2 @ A @ ( merge_list @ A @ As @ Ls2 ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As @ Ls2 ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_5294_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y3: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_5295_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_5296_revg__fun,axiom,
! [A: $tType] :
( ( revg @ A )
= ( ^ [A8: list @ A] : ( append @ A @ ( rev @ A @ A8 ) ) ) ) ).
% revg_fun
thf(fact_5297_revg_Osimps_I2_J,axiom,
! [A: $tType,A4: A,As: list @ A,B3: list @ A] :
( ( revg @ A @ ( cons @ A @ A4 @ As ) @ B3 )
= ( revg @ A @ As @ ( cons @ A @ A4 @ B3 ) ) ) ).
% revg.simps(2)
thf(fact_5298_revg_Osimps_I1_J,axiom,
! [A: $tType,B3: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B3 )
= B3 ) ).
% revg.simps(1)
thf(fact_5299_revg_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A6: A,As2: list @ A] :
( ( X
= ( cons @ A @ A6 @ As2 ) )
=> ( Y
!= ( revg @ A @ As2 @ ( cons @ A @ A6 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_5300_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,As2: list @ A] :
( ( X
= ( cons @ A @ A6 @ As2 ) )
=> ( ( Y
= ( revg @ A @ As2 @ ( cons @ A @ A6 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A6 @ As2 ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_5301_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys2: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys2 ) )
=> ( ! [Ys2: list @ A,Zs: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys2 != Zs )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Zs ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_5302_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
= ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
@ ^ [Xs3: list @ ( product_prod @ A @ B ),F4: A > ( option @ B )] : ( map_add @ A @ B @ F4 @ ( map_of @ A @ B @ Xs3 ) )
@ Xss
@ ^ [X2: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_5303_sorted__filter_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( filter2 @ A @ P @ L ) ) ) ) ).
% sorted_filter'
thf(fact_5304_last__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( P @ ( last @ A @ Xs ) )
=> ( ( last @ A @ ( filter2 @ A @ P @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last_filter
thf(fact_5305_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( concat @ A
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xs ) )
= ( concat @ A @ Xs ) ) ).
% concat_filter_neq_Nil
thf(fact_5306_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list @ A,Xs: list @ B,F2: A > C > C,B3: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X2: B] : F2 )
@ ( zip @ B @ A @ Xs @ Ys )
@ B3 )
= ( foldr @ A @ C @ F2 @ Ys @ B3 ) ) ) ).
% foldr_snd_zip
thf(fact_5307_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
( ( F2 @ Y )
= ( F2 @ X2 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y4: A,Z5: A] : Y4 = Z5
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_5308_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F2: A > B > B,Xs: list @ A,A4: B] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( ( foldr @ A @ B @ F2 @ Xs @ A4 )
= ( foldl @ B @ A
@ ^ [A8: B,B6: A] : ( F2 @ B6 @ A8 )
@ A4
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_5309_filter__eq__snocD,axiom,
! [A: $tType,P: A > $o,L: list @ A,L4: list @ A,X: A] :
( ( ( filter2 @ A @ P @ L )
= ( append @ A @ L4 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ( P @ X ) ) ) ).
% filter_eq_snocD
thf(fact_5310_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_5311_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_5312_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_5313_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_5314_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_5315_filter__nth__ex__nth,axiom,
! [A: $tType,N2: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) )
=> ? [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
& ( ord_less @ nat @ M3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter2 @ A @ P @ Xs ) @ N2 )
= ( nth @ A @ Xs @ M3 ) )
& ( ( filter2 @ A @ P @ ( take @ A @ M3 @ Xs ) )
= ( take @ A @ N2 @ ( filter2 @ A @ P @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_5316_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B6: A] : ( plus_plus @ A @ ( F4 @ X2 ) @ ( times_times @ A @ A8 @ B6 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_5317_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X2: A,Xs3: list @ A] :
( filter2 @ A
@ ^ [Y3: A] : Y3 != X2
@ ( rev @ A @ Xs3 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_5318_filter__rev__alt,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( ^ [P2: A > $o,L2: list @ A] : ( filter2 @ A @ P2 @ ( rev @ A @ L2 ) ) ) ) ).
% filter_rev_alt
thf(fact_5319_filter__rev__aux__alt,axiom,
! [A: $tType] :
( ( filter_rev_aux @ A )
= ( ^ [A8: list @ A,P2: A > $o,L2: list @ A] : ( append @ A @ ( filter2 @ A @ P2 @ ( rev @ A @ L2 ) ) @ A8 ) ) ) ).
% filter_rev_aux_alt
thf(fact_5320_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X2: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_5321_filter__rev__def,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% filter_rev_def
thf(fact_5322_remove__rev__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X2: A] :
( filter_rev @ A
@ ( comp @ $o @ $o @ A @ (~)
@ ( ^ [Y4: A,Z5: A] : Y4 = Z5
@ X2 ) ) ) ) ) ).
% remove_rev_def
thf(fact_5323_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X: B,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ X )
= ( none @ A ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P @ Xs ) @ X )
= ( none @ A ) ) ) ).
% map_of_None_filterD
thf(fact_5324_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_5325_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,A4: list @ A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter_rev_aux @ A @ A4 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ ( cons @ A @ X @ A4 ) @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( filter_rev_aux @ A @ A4 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ A4 @ P @ Xs ) ) ) ) ).
% filter_rev_aux.simps(2)
thf(fact_5326_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A4: list @ A,P: A > $o] :
( ( filter_rev_aux @ A @ A4 @ P @ ( nil @ A ) )
= A4 ) ).
% filter_rev_aux.simps(1)
thf(fact_5327_distinct__map__fst__filterI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),P: ( product_prod @ A @ B ) > $o] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( filter2 @ ( product_prod @ A @ B ) @ P @ Xs ) ) ) ) ).
% distinct_map_fst_filterI
thf(fact_5328_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X2: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P2 @ X2 ) @ ( cons @ A @ X2 @ Xt ) @ Xt )
@ Xs3
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_5329_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A4: nat] :
( ( foldr @ A @ nat
@ ^ [X2: A] : suc
@ L
@ A4 )
= ( plus_plus @ nat @ A4 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_5330_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_5331_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_5332_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_5333_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R3: A > B > $o] :
( relcompp @ ( list @ A ) @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( conversep @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_5334_zip__replicate,axiom,
! [A: $tType,B: $tType,I2: nat,X: A,J2: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I2 @ X ) @ ( replicate @ B @ J2 @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I2 @ J2 ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_5335_set__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_5336_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L4: list @ B,Q2: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ L @ L4 )
=> ( ( Q2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,X10: B,Ls: list @ A,Ls3: list @ B] :
( ( P @ X3 @ X10 )
=> ( ( list_all2 @ A @ B @ P @ Ls @ Ls3 )
=> ( ( Q2 @ Ls @ Ls3 )
=> ( Q2 @ ( cons @ A @ X3 @ Ls ) @ ( cons @ B @ X10 @ Ls3 ) ) ) ) )
=> ( Q2 @ L @ L4 ) ) ) ) ).
% Misc.list_all2_induct
thf(fact_5337_set__replicate__conv__if,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_5338_set__replicate__Suc,axiom,
! [A: $tType,N2: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_5339_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N2: nat,X: A] :
( ( replicate @ A @ ( suc @ N2 ) @ X )
= ( append @ A @ ( replicate @ A @ N2 @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_5340_zip__replicate1,axiom,
! [A: $tType,B: $tType,N2: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N2 @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N2 @ Ys ) ) ) ).
% zip_replicate1
thf(fact_5341_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_5342_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_5343_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N2: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N2 @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ Y )
@ ( take @ A @ N2 @ Xs ) ) ) ).
% zip_replicate2
thf(fact_5344_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A3: A > B > $o,B5: 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 @ B5 @ 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 @ B5 ) @ A3 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_5345_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R3: A > B > $o,A8: list @ A,B6: list @ B] :
? [Z3: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A8 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B6 ) ) ) ) ).
% list.in_rel
thf(fact_5346_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_5347_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_5348_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_5349_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_5350_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_5351_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_5352_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_5353_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_5354_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_5355_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_5356_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_5357_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
@ ^ [Y3: A] : ( X @ Y3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_5358_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R4: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R4 @ 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 @ R4 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R4 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R4 @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_5359_set__quicksort__by__rel,axiom,
! [A: $tType,R4: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( set2 @ A @ ( quicksort_by_rel @ A @ R4 @ Sl2 @ Xs ) )
= ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% set_quicksort_by_rel
thf(fact_5360_quicksort__by__rel__permutes,axiom,
! [A: $tType,R4: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( mset @ A @ ( quicksort_by_rel @ A @ R4 @ Sl2 @ Xs ) )
= ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% quicksort_by_rel_permutes
thf(fact_5361_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R4: A > A > $o,Sl2: list @ A] :
( ( quicksort_by_rel @ A @ R4 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ).
% quicksort_by_rel.simps(1)
thf(fact_5362_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list @ A,R4: A > A > $o,Xs: list @ A] :
( ( Sl2
!= ( nil @ A ) )
=> ( ( quicksort_by_rel @ A @ R4 @ Sl2 @ Xs )
= ( append @ A @ ( quicksort_by_rel @ A @ R4 @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% quicksort_by_rel_remove_acc_guared
thf(fact_5363_quicksort__by__rel__remove__acc,axiom,
! [A: $tType] :
( ( quicksort_by_rel @ A )
= ( ^ [R3: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R3 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% quicksort_by_rel_remove_acc
thf(fact_5364_sorted__wrt__quicksort__by__rel,axiom,
! [X14: $tType,R4: X14 > X14 > $o,Xs: list @ X14] :
( ! [X3: X14,Y2: X14] :
( ( R4 @ X3 @ Y2 )
| ( R4 @ Y2 @ X3 ) )
=> ( ! [X3: X14,Y2: X14,Z4: X14] :
( ( R4 @ X3 @ Y2 )
=> ( ( R4 @ Y2 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( sorted_wrt @ X14 @ R4 @ ( quicksort_by_rel @ X14 @ R4 @ ( nil @ X14 ) @ Xs ) ) ) ) ).
% sorted_wrt_quicksort_by_rel
thf(fact_5365_sorted__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) @ Xs ) ) ) ).
% sorted_quicksort_by_rel
thf(fact_5366_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R4: 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 ) ) @ R4 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R4 @ 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 @ R4 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R4 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R4 @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_5367_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
@ ^ [Y3: A] : ( X @ Y3 @ 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_5368_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A1: list @ A,A22: list @ A,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ A0 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A1 @ A22 ) ) )
=> ( ! [R8: A > A > $o,Sl: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R8 @ Sl @ ( nil @ A ) ) )
=> ( ! [R8: A > A > $o,Sl: list @ A,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 ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs4 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R8 @ Z3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R8 @ Sl @ Y6 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R8 @ Z3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs4 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R8 @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ R8 @ Sl @ Y6 ) ) @ Xb2 ) ) )
=> ( P @ R8 @ Sl @ ( cons @ A @ X3 @ Xs4 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_5369_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R4: 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 ) ) @ R4 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
=> ( ( quicksort_by_rel @ A @ R4 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
thf(fact_5370_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_5371_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( slice @ A @ From @ To @ Xs ) )
= ( nth @ A @ Xs @ From ) ) ) ) ).
% slice_head
thf(fact_5372_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( insert @ A @ X @ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% List.set_insert
thf(fact_5373_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_5374_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_5375_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_5376_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_5377_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X6: A] :
( ( member @ A @ X6 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X6 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_5378_sorted__hd__last,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( L
!= ( nil @ A ) )
=> ( ord_less_eq @ A @ ( hd @ A @ L ) @ ( last @ A @ L ) ) ) ) ) ).
% sorted_hd_last
thf(fact_5379_hd__last__singletonI,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( hd @ A @ Xs )
= ( last @ A @ Xs ) )
=> ( ( distinct @ A @ Xs )
=> ( Xs
= ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ) ).
% hd_last_singletonI
thf(fact_5380_hd__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( butlast @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_butlast
thf(fact_5381_rev__split__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( ( append @ A @ ( rev @ A @ ( tl @ A @ L ) ) @ ( cons @ A @ ( hd @ A @ L ) @ ( nil @ A ) ) )
= ( rev @ A @ L ) ) ) ).
% rev_split_conv
thf(fact_5382_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 )
@ ^ [K5: A,V4: C] : ( product_Pair @ A @ B @ K5 @ ( F2 @ V4 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_5383_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_5384_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_5385_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_5386_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E2: A] :
( Xs
!= ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_5387_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_5388_tl__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( last @ A @ ( tl @ A @ Xs ) ) ) ) ).
% tl_last
thf(fact_5389_rev__butlast__is__tl__rev,axiom,
! [A: $tType,L: list @ A] :
( ( rev @ A @ ( butlast @ A @ L ) )
= ( tl @ A @ ( rev @ A @ L ) ) ) ).
% rev_butlast_is_tl_rev
thf(fact_5390_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A3 ) ) ) ).
% tl_subset
thf(fact_5391_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( suc @ N2 ) ) ) ) ).
% Misc.nth_tl
thf(fact_5392_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,P: B > A > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ B @ Ys @ N3 ) @ ( nth @ A @ Xs @ N3 ) ) )
=> ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( P @ ( nth @ B @ ( tl @ B @ Ys ) @ N9 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N9 ) ) ) ) ) ).
% list_take_induct_tl2
thf(fact_5393_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( X
= ( hd @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_5394_butlast__rev__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ) ).
% butlast_rev_tl
thf(fact_5395_remove1__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
= ( tl @ A @ Xs ) ) ) ).
% remove1_tl
thf(fact_5396_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N: nat,L: list @ A,M2: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M2 @ N ) )
= ( drop @ A @ M2 @ ( take @ A @ N @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_5397_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R2: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X2: C] : R2
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R2 ) ) ) ).
% sum_list_triv
thf(fact_5398_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_5399_upt__0__eq__Nil__conv,axiom,
! [J2: nat] :
( ( ( upt @ ( zero_zero @ nat ) @ J2 )
= ( nil @ nat ) )
= ( J2
= ( zero_zero @ nat ) ) ) ).
% upt_0_eq_Nil_conv
thf(fact_5400_upt__merge,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I2 @ J2 )
& ( ord_less_eq @ nat @ J2 @ K ) )
=> ( ( append @ nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ K ) )
= ( upt @ I2 @ K ) ) ) ).
% upt_merge
thf(fact_5401_map__add__upt_H,axiom,
! [Ofs: nat,A4: nat,B3: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ Ofs )
@ ( upt @ A4 @ B3 ) )
= ( upt @ ( plus_plus @ nat @ A4 @ Ofs ) @ ( plus_plus @ nat @ B3 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_5402_upt__eq__append__conv,axiom,
! [I2: nat,J2: nat,Xs: list @ nat,Ys: list @ nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ( upt @ I2 @ J2 )
= ( append @ nat @ Xs @ Ys ) )
= ( ? [K5: nat] :
( ( ord_less_eq @ nat @ I2 @ K5 )
& ( ord_less_eq @ nat @ K5 @ J2 )
& ( ( upt @ I2 @ K5 )
= Xs )
& ( ( upt @ K5 @ J2 )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_5403_upt__append,axiom,
! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I2 ) @ ( upt @ I2 @ J2 ) )
= ( upt @ ( zero_zero @ nat ) @ J2 ) ) ) ).
% upt_append
thf(fact_5404_butlast__upt,axiom,
! [M: nat,N2: nat] :
( ( butlast @ nat @ ( upt @ M @ N2 ) )
= ( upt @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_5405_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M: nat,L: list @ A,N2: nat] :
( ( filter2 @ nat
@ ^ [I4: nat] : ( P @ ( nth @ A @ ( take @ A @ M @ L ) @ I4 ) )
@ ( upt @ N2 @ M ) )
= ( filter2 @ nat
@ ^ [I4: nat] : ( P @ ( nth @ A @ L @ I4 ) )
@ ( upt @ N2 @ M ) ) ) ).
% filter_upt_take_conv
thf(fact_5406_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_5407_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F2: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_5408_upt__eq__lel__conv,axiom,
! [L: nat,H2: nat,Is1: list @ nat,I2: nat,Is2: list @ nat] :
( ( ( upt @ L @ H2 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I2 @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I2 ) )
& ( Is2
= ( upt @ ( suc @ I2 ) @ H2 ) )
& ( ord_less_eq @ nat @ L @ I2 )
& ( ord_less @ nat @ I2 @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_5409_upt__filter__extend,axiom,
! [U: nat,U5: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U5 )
=> ( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ U @ I3 )
& ( ord_less @ nat @ I3 @ U5 ) )
=> ~ ( P @ I3 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_5410_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N2 @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_5411_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js: list @ nat,J2: nat,I2: nat] :
( ( ( filter2 @ nat
@ ^ [K5: nat] : ( P @ ( nth @ A @ L @ K5 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js @ ( cons @ nat @ J2 @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J2 @ I2 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% filter_upt_last
thf(fact_5412_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X4: set @ A,F2: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X4 )
=> ( ( finite_finite2 @ A @ X4 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F2 @ X2 ) )
@ X4 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_5413_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_5414_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_5415_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R3: A > B > $o] :
( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% option.rel_compp_Grp
thf(fact_5416_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R3: A > B > $o,A8: option @ A,B6: option @ B] :
? [Z3: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A8 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B6 ) ) ) ) ).
% option.in_rel
thf(fact_5417_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_5418_ordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A8: A,B6: A] : ( Less_eq @ B6 @ A8 )
@ ^ [A8: A,B6: A] : ( Less @ B6 @ A8 ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_dualI
thf(fact_5419_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B2: A] :
( ( Less_eq @ A6 @ B2 )
= ( ( Less @ A6 @ B2 )
| ( A6 = B2 ) ) )
=> ( ! [A6: A,B2: A] :
( ( Less @ A6 @ B2 )
=> ~ ( Less @ B2 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B2: A,C3: A] :
( ( Less @ A6 @ B2 )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ A6 @ C3 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_5420_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_5421_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_5422_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_5423_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_5424_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_5425_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_5426_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_5427_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_5428_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_5429_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_5430_less__length__takeWhile__conv,axiom,
! [A: $tType,I2: nat,P: A > $o,L: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( P @ ( nth @ A @ L @ J3 ) ) ) ) ) ).
% less_length_takeWhile_conv
thf(fact_5431_eq__len__takeWhile__conv,axiom,
! [A: $tType,I2: nat,P: A > $o,L: list @ A] :
( ( I2
= ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less_eq @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ( P @ ( nth @ A @ L @ J3 ) ) )
& ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% eq_len_takeWhile_conv
thf(fact_5432_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_finite2 @ 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_5433_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list @ A,P: B > $o,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( takeWhile @ B @ P @ Ys ) )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_snd
thf(fact_5434_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_5435_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_finite2 @ 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_5436_drop__takeWhile,axiom,
! [A: $tType,I2: nat,P: A > $o,L: list @ A] :
( ( ord_less_eq @ nat @ I2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
=> ( ( drop @ A @ I2 @ ( takeWhile @ A @ P @ L ) )
= ( takeWhile @ A @ P @ ( drop @ A @ I2 @ L ) ) ) ) ).
% drop_takeWhile
thf(fact_5437_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_finite2 @ 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_5438_folding__insort__key_Osorted__key__list__of__set__insert,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_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ 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 @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_5439_length__dropWhile__takeWhile,axiom,
! [A: $tType,X: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ X @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_dropWhile_takeWhile
thf(fact_5440_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2 )
= ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% sort_quicksort_by_rel
thf(fact_5441_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_5442_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_5443_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_5444_semigroup_Oassoc,axiom,
! [A: $tType,F2: A > A > A,A4: A,B3: A,C2: A] :
( ( semigroup @ A @ F2 )
=> ( ( F2 @ ( F2 @ A4 @ B3 ) @ C2 )
= ( F2 @ A4 @ ( F2 @ B3 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_5445_semigroup_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A,B2: A,C3: A] :
( ( F2 @ ( F2 @ A6 @ B2 ) @ C3 )
= ( F2 @ A6 @ ( F2 @ B2 @ C3 ) ) )
=> ( semigroup @ A @ F2 ) ) ).
% semigroup.intro
thf(fact_5446_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F4: A > A > A] :
! [A8: A,B6: A,C5: A] :
( ( F4 @ ( F4 @ A8 @ B6 ) @ C5 )
= ( F4 @ A8 @ ( F4 @ B6 @ C5 ) ) ) ) ) ).
% semigroup_def
thf(fact_5447_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_5448_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_5449_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_5450_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_5451_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2 )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_5452_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F4: A > A > A,Z3: A,Inverse2: A > A] :
( ( semigroup @ A @ F4 )
& ( group_axioms @ A @ F4 @ Z3 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_5453_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F4: A > A > A,Z3: A] :
( ( semigroup @ A @ F4 )
& ( monoid_axioms @ A @ F4 @ Z3 ) ) ) ) ).
% monoid_def
thf(fact_5454_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_5455_insert__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( insert2 @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_5456_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F4: A > A > A,Z3: A] :
( ! [A8: A] :
( ( F4 @ Z3 @ A8 )
= A8 )
& ! [A8: A] :
( ( F4 @ A8 @ Z3 )
= A8 ) ) ) ) ).
% monoid_axioms_def
thf(fact_5457_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_5458_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_5459_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2 ) )
= ( mergesort_remdups @ A ) ) ) ).
% remdup_sort_mergesort_remdups
thf(fact_5460_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_5461_lazI,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ B3 ) )
=> ( P @ ( nth @ A @ A4 @ I3 ) @ ( nth @ B @ B3 @ I3 ) ) )
=> ( list_all_zip @ A @ B @ P @ A4 @ B3 ) ) ) ).
% lazI
thf(fact_5462_laz__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q2: A > B > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [X2: A,Y3: B] :
( ( P @ X2 @ Y3 )
& ( Q2 @ X2 @ Y3 ) )
@ A4
@ B3 )
= ( ( list_all_zip @ A @ B @ P @ A4 @ B3 )
& ( list_all_zip @ A @ B @ Q2 @ A4 @ B3 ) ) ) ).
% laz_conj
thf(fact_5463_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: A > $o,A3: list @ A,B5: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A8: A,B6: B] : ( P @ A8 )
@ A3
@ B5 )
= ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B5 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ A3 ) )
=> ( P @ X2 ) ) ) ) ).
% laz_weak_Pa
thf(fact_5464_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: B > $o,A3: list @ A,B5: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A8: A] : P
@ A3
@ B5 )
= ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B5 ) )
& ! [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ B5 ) )
=> ( P @ X2 ) ) ) ) ).
% laz_weak_Pb
thf(fact_5465_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A3: list @ A,B5: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A8: A,B6: B] :
? [X7: C] : ( P @ A8 @ B6 @ X7 )
@ A3
@ B5 )
=> ~ ! [C7: list @ C] :
( ( list_all_zip @ A @ C
@ ^ [A8: A,C5: C] :
? [B6: B] : ( P @ A8 @ B6 @ C5 )
@ A3
@ C7 )
=> ~ ( list_all_zip @ B @ C
@ ^ [B6: B,C5: C] :
? [A8: A] : ( P @ A8 @ B6 @ C5 )
@ B5
@ C7 ) ) ) ).
% laz_swap_ex
thf(fact_5466_laz__eq,axiom,
! [A: $tType,A4: list @ A,B3: list @ A] :
( ( list_all_zip @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ A4
@ B3 )
= ( A4 = B3 ) ) ).
% laz_eq
thf(fact_5467_laz__len,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B @ P @ A4 @ B3 )
=> ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) ) ) ).
% laz_len
thf(fact_5468_list__all__zip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o] : ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) ) ).
% list_all_zip.simps(1)
thf(fact_5469_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A4: A,As: list @ A,B3: B,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) )
= ( ( P @ A4 @ B3 )
& ( list_all_zip @ A @ B @ P @ As @ Bs ) ) ) ).
% list_all_zip.simps(2)
thf(fact_5470_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F2: C > A,As: list @ C,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F2 @ As ) @ Bs )
= ( list_all_zip @ C @ B
@ ^ [A8: C] : ( P @ ( F2 @ A8 ) )
@ As
@ Bs ) ) ).
% list_all_zip_map1
thf(fact_5471_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As: list @ A,F2: C > B,Bs: list @ C] :
( ( list_all_zip @ A @ B @ P @ As @ ( map @ C @ B @ F2 @ Bs ) )
= ( list_all_zip @ A @ C
@ ^ [A8: A,B6: C] : ( P @ A8 @ ( F2 @ B6 ) )
@ As
@ Bs ) ) ).
% list_all_zip_map2
thf(fact_5472_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V: A,Va2: list @ A] :
~ ( list_all_zip @ A @ B @ P @ ( cons @ A @ V @ Va2 ) @ ( nil @ B ) ) ).
% list_all_zip.simps(3)
thf(fact_5473_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V: B,Va2: list @ B] :
~ ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ V @ Va2 ) ) ).
% list_all_zip.simps(4)
thf(fact_5474_list__all__zip_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ Y ) )
=> ( ! [A6: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ Bs2 ) )
=> ( Y
= ( ~ ( ( X @ A6 @ B2 )
& ( list_all_zip @ A @ B @ X @ As2 @ Bs2 ) ) ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> Y ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V3: B,Va: list @ B] :
( Xb
= ( cons @ B @ V3 @ Va ) )
=> Y ) ) ) ) ) ) ).
% list_all_zip.elims(1)
thf(fact_5475_list__all__zip_Oelims_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 )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ! [A6: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ Bs2 ) )
=> ~ ( ( X @ A6 @ B2 )
& ( list_all_zip @ A @ B @ X @ As2 @ Bs2 ) ) ) ) ) ) ).
% list_all_zip.elims(2)
thf(fact_5476_list__all__zip_Oelims_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 )
=> ( ! [A6: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ Bs2 ) )
=> ( ( X @ A6 @ B2 )
& ( list_all_zip @ A @ B @ X @ As2 @ Bs2 ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( Xb
!= ( cons @ B @ V3 @ Va ) ) ) ) ) ) ).
% list_all_zip.elims(3)
thf(fact_5477_length__remdups__card,axiom,
! [A: $tType,L: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ L ) )
= ( finite_card @ A @ ( set2 @ A @ L ) ) ) ).
% length_remdups_card
thf(fact_5478_list__all__zip__alt,axiom,
! [B: $tType,A: $tType] :
( ( list_all_zip @ A @ B )
= ( ^ [P2: A > B > $o,As3: list @ A,Bs3: list @ B] :
( ( ( size_size @ ( list @ A ) @ As3 )
= ( size_size @ ( list @ B ) @ Bs3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ As3 ) )
=> ( P2 @ ( nth @ A @ As3 @ I4 ) @ ( nth @ B @ Bs3 @ I4 ) ) ) ) ) ) ).
% list_all_zip_alt
thf(fact_5479_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,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ 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 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) ) ) )
=> ( ( X @ A6 @ B2 )
& ( list_all_zip @ A @ B @ X @ As2 @ 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_5480_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,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ 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 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) ) ) )
=> ~ ( ( X @ A6 @ B2 )
& ( list_all_zip @ A @ B @ X @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
thf(fact_5481_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,As2: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As2 ) )
=> ! [B2: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B2 @ Bs2 ) )
=> ( ( Y
= ( ( X @ A6 @ B2 )
& ( list_all_zip @ A @ B @ X @ As2 @ 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 @ As2 ) @ ( cons @ B @ B2 @ 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_5482_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_5483_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F2: A > B,L: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( map @ A @ B @ F2 @ L ) )
= L ) ) ).
% inj_map_inv_f
thf(fact_5484_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_5485_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B2: A] :
( ( Less @ A6 @ B2 )
= ( ( Less_eq @ A6 @ B2 )
& ( A6 != B2 ) ) )
=> ( ! [A6: A,B2: A] :
( ( Less_eq @ A6 @ B2 )
=> ( ( Less_eq @ B2 @ A6 )
=> ( A6 = B2 ) ) )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% ordering_axioms.intro
thf(fact_5486_ordering__axioms__def,axiom,
! [A: $tType] :
( ( ordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A8: A,B6: A] :
( ( Less2 @ A8 @ B6 )
= ( ( Less_eq2 @ A8 @ B6 )
& ( A8 != B6 ) ) )
& ! [A8: A,B6: A] :
( ( Less_eq2 @ A8 @ B6 )
=> ( ( Less_eq2 @ B6 @ A8 )
=> ( A8 = B6 ) ) ) ) ) ) ).
% ordering_axioms_def
thf(fact_5487_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_5488_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_5489_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_5490_partial__preordering__def,axiom,
! [A: $tType] :
( ( partial_preordering @ A )
= ( ^ [Less_eq2: A > A > $o] :
( ! [A8: A] : ( Less_eq2 @ A8 @ A8 )
& ! [A8: A,B6: A,C5: A] :
( ( Less_eq2 @ A8 @ B6 )
=> ( ( Less_eq2 @ B6 @ C5 )
=> ( Less_eq2 @ A8 @ C5 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_5491_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,A4: A,B3: A,C2: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less_eq @ B3 @ C2 )
=> ( Less_eq @ A4 @ C2 ) ) ) ) ).
% partial_preordering.trans
thf(fact_5492_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o] :
( ! [A6: A] : ( Less_eq @ A6 @ A6 )
=> ( ! [A6: A,B2: A,C3: A] :
( ( Less_eq @ A6 @ B2 )
=> ( ( Less_eq @ B2 @ C3 )
=> ( Less_eq @ A6 @ C3 ) ) )
=> ( partial_preordering @ A @ Less_eq ) ) ) ).
% partial_preordering.intro
thf(fact_5493_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_5494_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% order.partial_preordering_axioms
thf(fact_5495_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_5496_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_5497_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_5498_sym__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( sym @ B @ ( R2 @ X3 ) ) )
=> ( sym @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% sym_INTER
thf(fact_5499_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_5500_sym__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ ( converse @ A @ A @ R2 ) )
= ( sym @ A @ R2 ) ) ).
% sym_converse
thf(fact_5501_symD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B3: A,A4: A] :
( ( sym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ).
% symD
thf(fact_5502_symE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B3: A,A4: A] :
( ( sym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ).
% symE
thf(fact_5503_symI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A6: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A6 ) @ R2 ) )
=> ( sym @ A @ R2 ) ) ).
% symI
thf(fact_5504_sym__def,axiom,
! [A: $tType] :
( ( sym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R5 ) ) ) ) ).
% sym_def
thf(fact_5505_sym__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R2 )
=> ( ( sym @ A @ S2 )
=> ( sym @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% sym_Int
thf(fact_5506_sym__inv__image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),F2: B > A] :
( ( sym @ A @ R2 )
=> ( sym @ B @ ( inv_image @ A @ B @ R2 @ F2 ) ) ) ).
% sym_inv_image
thf(fact_5507_sym__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( sym @ A @ ( id_on @ A @ A3 ) ) ).
% sym_Id_on
thf(fact_5508_sym__Id,axiom,
! [A: $tType] : ( sym @ A @ ( id2 @ A ) ) ).
% sym_Id
thf(fact_5509_sym__conv__converse__eq,axiom,
! [A: $tType] :
( ( sym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( ( converse @ A @ A @ R5 )
= R5 ) ) ) ).
% sym_conv_converse_eq
thf(fact_5510_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_5511_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F4: A > A > A,Z3: A] :
! [A8: A] :
( ( F4 @ A8 @ Z3 )
= A8 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_5512_sym__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R2 )
=> ( ( sym @ A @ S2 )
=> ( sym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% sym_Un
thf(fact_5513_sym__Int__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( sym @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( converse @ A @ A @ R2 ) ) ) ).
% sym_Int_converse
thf(fact_5514_sym__Un__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( sym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( converse @ A @ A @ R2 ) ) ) ).
% sym_Un_converse
thf(fact_5515_sym__UNION,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( sym @ B @ ( R2 @ X3 ) ) )
=> ( sym @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% sym_UNION
thf(fact_5516_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F4: A > A > A,Z3: A] :
( ( abel_semigroup @ A @ F4 )
& ( comm_monoid_axioms @ A @ F4 @ Z3 ) ) ) ) ).
% comm_monoid_def
thf(fact_5517_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_5518_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_5519_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_5520_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_5521_preordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A
@ ^ [A8: A,B6: A] : ( Less_eq @ B6 @ A8 )
@ ^ [A8: A,B6: A] : ( Less @ B6 @ A8 ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ).
% preordering_dualI
thf(fact_5522_preordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B2: A] :
( ( Less_eq @ A6 @ B2 )
= ( ( Less @ A6 @ B2 )
| ( A6 = B2 ) ) )
=> ( ! [A6: A,B2: A] :
( ( Less @ A6 @ B2 )
=> ~ ( Less @ B2 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B2: A,C3: A] :
( ( Less @ A6 @ B2 )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ A6 @ C3 ) ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_5523_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_5524_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_5525_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( ( Less_eq @ B3 @ C2 )
=> ( Less @ A4 @ C2 ) ) ) ) ).
% preordering.strict_trans2
thf(fact_5526_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less @ B3 @ C2 )
=> ( Less @ A4 @ C2 ) ) ) ) ).
% preordering.strict_trans1
thf(fact_5527_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( ( Less @ B3 @ C2 )
=> ( Less @ A4 @ C2 ) ) ) ) ).
% preordering.strict_trans
thf(fact_5528_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_5529_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_5530_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F2: A > A > A,B3: A,A4: A,C2: A] :
( ( abel_semigroup @ A @ F2 )
=> ( ( F2 @ B3 @ ( F2 @ A4 @ C2 ) )
= ( F2 @ A4 @ ( F2 @ B3 @ C2 ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_5531_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_5532_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_5533_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_5534_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_5535_order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.preordering_axioms
thf(fact_5536_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_5537_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_5538_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_5539_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A,B2: A] :
( ( F2 @ A6 @ B2 )
= ( F2 @ B2 @ A6 ) )
=> ( abel_s757365448890700780axioms @ A @ F2 ) ) ).
% abel_semigroup_axioms.intro
thf(fact_5540_abel__semigroup__axioms__def,axiom,
! [A: $tType] :
( ( abel_s757365448890700780axioms @ A )
= ( ^ [F4: A > A > A] :
! [A8: A,B6: A] :
( ( F4 @ A8 @ B6 )
= ( F4 @ B6 @ A8 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_5541_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B2: A] :
( ( Less @ A6 @ B2 )
= ( ( Less_eq @ A6 @ B2 )
& ~ ( Less_eq @ B2 @ A6 ) ) )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering_axioms.intro
thf(fact_5542_preordering__axioms__def,axiom,
! [A: $tType] :
( ( preordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
! [A8: A,B6: A] :
( ( Less2 @ A8 @ B6 )
= ( ( Less_eq2 @ A8 @ B6 )
& ~ ( Less_eq2 @ B6 @ A8 ) ) ) ) ) ).
% preordering_axioms_def
thf(fact_5543_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_5544_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_5545_abel__semigroup__def,axiom,
! [A: $tType] :
( ( abel_semigroup @ A )
= ( ^ [F4: A > A > A] :
( ( semigroup @ A @ F4 )
& ( abel_s757365448890700780axioms @ A @ F4 ) ) ) ) ).
% abel_semigroup_def
thf(fact_5546_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_5547_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A5: set @ A,F4: ( product_prod @ A @ B ) > C,A8: A] :
( if @ ( B > C ) @ ( member @ A @ A8 @ A5 )
@ ^ [B6: B] : ( F4 @ ( product_Pair @ A @ B @ A8 @ B6 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_5548_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_5549_comm__monoid__set_Oinsert_H,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,I: set @ B,P3: B > A,I2: B] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I )
& ( ( P3 @ X2 )
!= Z2 ) ) ) )
=> ( ( ( member @ B @ I2 @ I )
=> ( ( groups_comm_monoid_G @ A @ B @ F2 @ Z2 @ P3 @ ( insert2 @ B @ I2 @ I ) )
= ( groups_comm_monoid_G @ A @ B @ F2 @ Z2 @ P3 @ I ) ) )
& ( ~ ( member @ B @ I2 @ I )
=> ( ( groups_comm_monoid_G @ A @ B @ F2 @ Z2 @ P3 @ ( insert2 @ B @ I2 @ I ) )
= ( F2 @ ( P3 @ I2 ) @ ( groups_comm_monoid_G @ A @ B @ F2 @ Z2 @ P3 @ I ) ) ) ) ) ) ) ).
% comm_monoid_set.insert'
thf(fact_5550_find__SomeD_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_5551_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ! [X3: nat] :
( ( member @ nat @ X3 @ As2 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ).
% in_range.pelims(3)
thf(fact_5552_find__SomeD_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( P @ X ) ) ).
% find_SomeD(1)
thf(fact_5553_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_5554_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F2: A > A > A,Z2: A,P3: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_G @ A @ B @ F2 @ Z2 @ P3 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty'
thf(fact_5555_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( Y
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ As2 )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ) ) ).
% in_range.pelims(1)
thf(fact_5556_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ~ ! [X6: nat] :
( ( member @ nat @ X6 @ As2 )
=> ( ord_less @ nat @ X6 @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ).
% in_range.pelims(2)
thf(fact_5557_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( As2
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
thf(fact_5558_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( As2
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
thf(fact_5559_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 )
=> ~ ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( Y
= ( As2
= ( 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 ) @ H @ As2 ) ) ) ) ) ) ).
% one_assn_raw.pelims(1)
thf(fact_5560_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F2: A > A > A,Z2: A,G: nat > A,N2: nat] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2
@ ^ [I4: nat] : ( F2 @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% comm_monoid_set.in_pairs_0
thf(fact_5561_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F2: A > A > A,Z2: A,G: nat > A,M: nat,N2: nat] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2
@ ^ [I4: nat] : ( F2 @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% comm_monoid_set.in_pairs
thf(fact_5562_comm__monoid__set_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > A > A,Z2: A,I: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ I )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I )
=> ( finite_finite2 @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F2 @ Z2 @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I ) ) )
= ( groups_comm_monoid_F @ A @ B @ F2 @ Z2
@ ^ [X2: B] : ( groups_comm_monoid_F @ A @ C @ F2 @ Z2 @ G @ ( A3 @ X2 ) )
@ I ) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
thf(fact_5563_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F2: A > A > A,Z2: A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty
thf(fact_5564_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% comm_monoid_mult_class.prod_def
thf(fact_5565_comm__monoid__set_Oinsert,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,X: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( insert2 @ B @ X @ A3 ) )
= ( F2 @ ( G @ X ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ A3 ) ) ) ) ) ) ).
% comm_monoid_set.insert
thf(fact_5566_comm__monoid__set_Oinsert__if,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,X: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( insert2 @ B @ X @ A3 ) )
= ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( insert2 @ B @ X @ A3 ) )
= ( F2 @ ( G @ X ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ A3 ) ) ) ) ) ) ) ).
% comm_monoid_set.insert_if
thf(fact_5567_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,C6: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C6 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C6 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C6 ) )
= ( 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 ) @ G @ C6 ) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
thf(fact_5568_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,G: B > A,X: B] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( insert2 @ B @ X @ A3 ) )
= ( F2 @ ( G @ X ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.insert_remove
thf(fact_5569_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,X: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ A3 )
= ( F2 @ ( G @ X ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.remove
thf(fact_5570_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,A3: set @ B,B5: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( F2 @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ A3 ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ G @ B5 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
thf(fact_5571_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F2: A > A > A,Z2: A,S: set @ B,A4: B,B3: B > A,C2: B > A] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( C2 @ K5 ) )
@ S )
= ( F2 @ ( B3 @ A4 ) @ ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F2 @ Z2
@ ^ [K5: B] : ( if @ A @ ( K5 = A4 ) @ ( B3 @ K5 ) @ ( C2 @ K5 ) )
@ S )
= ( groups_comm_monoid_F @ A @ B @ F2 @ Z2 @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.delta_remove
thf(fact_5572_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F2: A > A > A,Z2: A,G: nat > A,K: nat,N2: nat] :
( ( groups778175481326437816id_set @ A @ F2 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2
@ ^ [M4: nat] : ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M4 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F2 @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% comm_monoid_set.nat_group
thf(fact_5573_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,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y3 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V4 ) @ ( times_times @ nat @ Y3 @ U3 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_5574_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R5: A > A > $o,X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_5575_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_5576_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: ( 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 ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Bx @ By )
=> ( ! [A6: A,B2: B,Aa2: A,Ba: B] :
( ( R2 @ ( product_Pair @ A @ B @ A6 @ B2 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R2 @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A6 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_5577_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R2: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R2 @ ( 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,B2: B] :
( ( R2 @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R2 @ ( product_Pair @ A @ B @ A6 @ B2 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_5578_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: ( 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 ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A6: A,B2: B,Aa2: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( ( R2 @ ( product_Pair @ A @ B @ A6 @ B2 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A6 @ B2 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_5579_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_5580_next_Osimps,axiom,
! [V: code_natural,W: code_natural] :
( ( next @ ( product_Pair @ code_natural @ code_natural @ V @ W ) )
= ( 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 @ W @ ( 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 @ W @ ( 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 @ W @ ( 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 @ W @ ( 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_5581_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R5: set @ ( product_prod @ A @ A ),F4: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu: product_prod @ A2 @ A2] :
? [A8: A,B6: A] :
( ( Uu
= ( product_Pair @ A2 @ A2 @ ( F4 @ A8 ) @ ( F4 @ B6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R5 ) ) ) ) ) ).
% dir_image_def
thf(fact_5582_accp__acc__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( accp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( acc @ A @ R2 ) ) ) ) ).
% accp_acc_eq
thf(fact_5583_acc__induct__rule,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A4 @ ( acc @ A @ R2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( acc @ A @ R2 ) )
=> ( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A4 ) ) ) ).
% acc_induct_rule
thf(fact_5584_not__acc__down,axiom,
! [A: $tType,X: A,R4: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ ( acc @ A @ R4 ) )
=> ~ ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X ) @ R4 )
=> ( member @ A @ Z4 @ ( acc @ A @ R4 ) ) ) ) ).
% not_acc_down
thf(fact_5585_acc__downward,axiom,
! [A: $tType,B3: A,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( member @ A @ B3 @ ( acc @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( member @ A @ A4 @ ( acc @ A @ R2 ) ) ) ) ).
% acc_downward
thf(fact_5586_acc__induct,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A4 @ ( acc @ A @ R2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( acc @ A @ R2 ) )
=> ( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A4 ) ) ) ).
% acc_induct
thf(fact_5587_acc_Ointros,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R2 )
=> ( member @ A @ Y2 @ ( acc @ A @ R2 ) ) )
=> ( member @ A @ X @ ( acc @ A @ R2 ) ) ) ).
% acc.intros
thf(fact_5588_acc_Osimps,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A4 @ ( acc @ A @ R2 ) )
= ( ? [X2: A] :
( ( A4 = X2 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R2 )
=> ( member @ A @ Y3 @ ( acc @ A @ R2 ) ) ) ) ) ) ).
% acc.simps
thf(fact_5589_acc_Ocases,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A4 @ ( acc @ A @ R2 ) )
=> ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ A4 ) @ R2 )
=> ( member @ A @ Y6 @ ( acc @ A @ R2 ) ) ) ) ).
% acc.cases
thf(fact_5590_acc__subset__induct,axiom,
! [A: $tType,D4: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ D4 @ ( acc @ A @ R4 ) )
=> ( ! [X3: A,Z4: A] :
( ( member @ A @ X3 @ D4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X3 ) @ R4 )
=> ( member @ A @ Z4 @ D4 ) ) )
=> ( ( member @ A @ X @ D4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ D4 )
=> ( ! [Z7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z7 @ X3 ) @ R4 )
=> ( P @ Z7 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_5591_acc__downwards,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A ),B3: A] :
( ( member @ A @ A4 @ ( acc @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ A @ B3 @ ( acc @ A @ R2 ) ) ) ) ).
% acc_downwards
thf(fact_5592_acc__downwards__aux,axiom,
! [A: $tType,B3: A,A4: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ A @ A4 @ ( acc @ A @ R2 ) )
=> ( member @ A @ B3 @ ( acc @ A @ R2 ) ) ) ) ).
% acc_downwards_aux
thf(fact_5593_acc__def,axiom,
! [A: $tType] :
( ( acc @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ( accp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 ) ) ) ) ) ).
% acc_def
thf(fact_5594_Random_Orange__def,axiom,
( range
= ( ^ [K5: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K5 )
@ ^ [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
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V4 @ ( 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 ) )
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V4 @ K5 ) ) ) ) ) ).
% Random.range_def
thf(fact_5595_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_5596_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K5: code_natural,F4: B > A > ( product_prod @ B @ A ),X2: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K5
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X2 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F4 @ X2 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K5 @ ( one_one @ code_natural ) ) @ F4 ) ) ) ) ) ).
% iterate.simps
thf(fact_5597_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_5598_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_5599_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A11: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A11 )
=> ( ( A11
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A11 )
=> ( P @ X6 ) )
=> ( P @ ( Lub @ A11 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_5600_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P2: A > $o] :
! [A5: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( P2 @ X2 ) )
=> ( P2 @ ( Lub2 @ A5 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_5601_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_5602_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_5603_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_5604_unit__abs__eta__conv,axiom,
! [A: $tType,F2: product_unit > A] :
( ( ^ [U3: product_unit] : ( F2 @ product_Unity ) )
= F2 ) ).
% unit_abs_eta_conv
thf(fact_5605_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_5606_sup__unit__def,axiom,
( ( sup_sup @ product_unit )
= ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_5607_fixp__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A,G: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
=> ( ! [Z10: A] : ( ord_less_eq @ A @ ( F2 @ Z10 ) @ ( G @ Z10 ) )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F2 ) @ ( comple115746919287870866o_fixp @ A @ G ) ) ) ) ) ) ).
% fixp_mono
thf(fact_5608_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
thf(fact_5609_Inf__unit__def,axiom,
( ( complete_Inf_Inf @ product_unit )
= ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_5610_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu3: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_5611_top__unit__def,axiom,
( ( top_top @ product_unit )
= product_Unity ) ).
% top_unit_def
thf(fact_5612_Unity__def,axiom,
( product_Unity
= ( product_Abs_unit @ $true ) ) ).
% Unity_def
thf(fact_5613_Sup__unit__def,axiom,
( ( complete_Sup_Sup @ product_unit )
= ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_5614_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_5615_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_5616_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_5617_abstract__filter__def,axiom,
! [A: $tType] :
( ( abstract_filter @ A )
= ( ^ [F4: product_unit > ( filter @ A )] : ( F4 @ product_Unity ) ) ) ).
% abstract_filter_def
thf(fact_5618_CODE__ABORT__def,axiom,
! [A: $tType] :
( ( cODE_ABORT @ A )
= ( ^ [F4: product_unit > A] : ( F4 @ product_Unity ) ) ) ).
% CODE_ABORT_def
thf(fact_5619_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_5620_old_Ounit_Ocase,axiom,
! [A: $tType,F2: A] :
( ( product_case_unit @ A @ F2 @ product_Unity )
= F2 ) ).
% old.unit.case
thf(fact_5621_scomp__fcomp,axiom,
! [A: $tType,C: $tType,B: $tType,E: $tType,D: $tType,F2: A > ( product_prod @ D @ E ),G: D > E > C,H2: C > B] :
( ( fcomp @ A @ C @ B @ ( product_scomp @ A @ D @ E @ C @ F2 @ G ) @ H2 )
= ( product_scomp @ A @ D @ E @ B @ F2
@ ^ [X2: D] : ( fcomp @ E @ C @ B @ ( G @ X2 ) @ H2 ) ) ) ).
% scomp_fcomp
thf(fact_5622_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H2: A > B,F2: A,Unit: product_unit] :
( ( H2 @ ( product_case_unit @ A @ F2 @ Unit ) )
= ( product_case_unit @ B @ ( H2 @ F2 ) @ Unit ) ) ).
% unit.case_distrib
thf(fact_5623_fcomp__scomp,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F2: A > E,G: E > ( product_prod @ C @ D ),H2: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( fcomp @ A @ E @ ( product_prod @ C @ D ) @ F2 @ G ) @ H2 )
= ( fcomp @ A @ E @ B @ F2 @ ( product_scomp @ E @ C @ D @ B @ G @ H2 ) ) ) ).
% fcomp_scomp
thf(fact_5624_symp__INF,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > B > B > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( symp @ B @ ( R2 @ X3 ) ) )
=> ( symp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ S ) ) ) ) ).
% symp_INF
thf(fact_5625_times__integer__code_I3_J,axiom,
! [M: num,N2: num] :
( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N2 ) )
= ( code_Pos @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_integer_code(3)
thf(fact_5626_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_5627_sympD,axiom,
! [A: $tType,R2: A > A > $o,B3: A,A4: A] :
( ( symp @ A @ R2 )
=> ( ( R2 @ B3 @ A4 )
=> ( R2 @ A4 @ B3 ) ) ) ).
% sympD
thf(fact_5628_sympE,axiom,
! [A: $tType,R2: A > A > $o,B3: A,A4: A] :
( ( symp @ A @ R2 )
=> ( ( R2 @ B3 @ A4 )
=> ( R2 @ A4 @ B3 ) ) ) ).
% sympE
thf(fact_5629_sympI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [A6: A,B2: A] :
( ( R2 @ A6 @ B2 )
=> ( R2 @ B2 @ A6 ) )
=> ( symp @ A @ R2 ) ) ).
% sympI
thf(fact_5630_symp__def,axiom,
! [A: $tType] :
( ( symp @ A )
= ( ^ [R5: A > A > $o] :
! [X2: A,Y3: A] :
( ( R5 @ X2 @ Y3 )
=> ( R5 @ Y3 @ X2 ) ) ) ) ).
% symp_def
thf(fact_5631_symp__sup,axiom,
! [A: $tType,R2: A > A > $o,S2: A > A > $o] :
( ( symp @ A @ R2 )
=> ( ( symp @ A @ S2 )
=> ( symp @ A @ ( sup_sup @ ( A > A > $o ) @ R2 @ S2 ) ) ) ) ).
% symp_sup
thf(fact_5632_symp__inf,axiom,
! [A: $tType,R2: A > A > $o,S2: A > A > $o] :
( ( symp @ A @ R2 )
=> ( ( symp @ A @ S2 )
=> ( symp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S2 ) ) ) ) ).
% symp_inf
thf(fact_5633_symp__conversep,axiom,
! [A: $tType] :
( ( symp @ A )
= ( ^ [R3: A > A > $o] : ( ord_less_eq @ ( A > A > $o ) @ ( conversep @ A @ A @ R3 ) @ R3 ) ) ) ).
% symp_conversep
thf(fact_5634_symp__sym__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( symp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( sym @ A @ R2 ) ) ).
% symp_sym_eq
thf(fact_5635_symp__SUP,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > B > B > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( symp @ B @ ( R2 @ X3 ) ) )
=> ( symp @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ S ) ) ) ) ).
% symp_SUP
thf(fact_5636_compat__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_compat @ A @ A2 )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 ),F4: A > A2] :
! [A8: A,B6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R5 )
=> ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F4 @ A8 ) @ ( F4 @ B6 ) ) @ R9 ) ) ) ) ).
% compat_def
thf(fact_5637_times__integer__code_I6_J,axiom,
! [M: num,N2: num] :
( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N2 ) )
= ( code_Pos @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_integer_code(6)
thf(fact_5638_times__integer__code_I5_J,axiom,
! [M: num,N2: num] :
( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N2 ) )
= ( code_Neg @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_integer_code(5)
thf(fact_5639_times__integer__code_I4_J,axiom,
! [M: num,N2: num] :
( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N2 ) )
= ( code_Neg @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_integer_code(4)
thf(fact_5640_iso__backward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R7: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F2: B > A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R7 )
=> ( ( bNF_Wellorder_iso @ B @ A @ R2 @ R7 @ F2 )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R2 ) @ F2 @ X ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R2 ) @ F2 @ Y ) ) @ R2 ) ) ) ).
% iso_backward
thf(fact_5641_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( bij_betw @ A @ B @ F4 @ ( field2 @ A @ R5 ) @ ( field2 @ B @ R9 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R5 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( field2 @ A @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R5 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X2 ) @ ( F4 @ Y3 ) ) @ R9 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_5642_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_5643_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R2 @ R7 @ F2 )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R7 ) ) ) ).
% iso_forward
thf(fact_5644_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa ) )
= ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% times_natural.rep_eq
thf(fact_5645_sndsp_Ocases,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,A4: B] :
( ( basic_sndsp @ A @ B @ P3 @ A4 )
=> ( A4
= ( product_snd @ A @ B @ P3 ) ) ) ).
% sndsp.cases
thf(fact_5646_sndsp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_sndsp @ A @ B )
= ( ^ [P5: product_prod @ A @ B,A8: B] :
( A8
= ( product_snd @ A @ B @ P5 ) ) ) ) ).
% sndsp.simps
thf(fact_5647_sndsp_Ointros,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] : ( basic_sndsp @ A @ B @ P3 @ ( product_snd @ A @ B @ P3 ) ) ).
% sndsp.intros
thf(fact_5648_transp__INF,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > B > B > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( transp @ B @ ( R2 @ X3 ) ) )
=> ( transp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ S ) ) ) ) ).
% transp_INF
thf(fact_5649_pred__prod__beta,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P2: A > $o,Q: B > $o,Xy: product_prod @ A @ B] :
( ( P2 @ ( product_fst @ A @ B @ Xy ) )
& ( Q @ ( product_snd @ A @ B @ Xy ) ) ) ) ) ).
% pred_prod_beta
thf(fact_5650_times__int__code_I3_J,axiom,
! [M: num,N2: num] :
( ( times_times @ int @ ( pos @ M ) @ ( pos @ N2 ) )
= ( pos @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_int_code(3)
thf(fact_5651_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P1: A > $o,P25: B > $o,A4: A,B3: B] :
( ( basic_pred_prod @ A @ B @ P1 @ P25 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( P1 @ A4 )
& ( P25 @ B3 ) ) ) ).
% pred_prod_inject
thf(fact_5652_transpD,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z2: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X @ Z2 ) ) ) ) ).
% transpD
thf(fact_5653_transpE,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z2: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X @ Z2 ) ) ) ) ).
% transpE
thf(fact_5654_transpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A,Y2: A,Z4: A] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( transp @ A @ R2 ) ) ).
% transpI
thf(fact_5655_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] :
! [X2: A,Y3: A,Z3: A] :
( ( R5 @ X2 @ Y3 )
=> ( ( R5 @ Y3 @ Z3 )
=> ( R5 @ X2 @ Z3 ) ) ) ) ) ).
% transp_def
thf(fact_5656_transp__equality,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 ) ).
% transp_equality
thf(fact_5657_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_5658_transp__singleton,axiom,
! [A: $tType,A4: A] :
( transp @ A
@ ^ [X2: A,Y3: A] :
( ( X2 = A4 )
& ( Y3 = A4 ) ) ) ).
% transp_singleton
thf(fact_5659_transp__empty,axiom,
! [A: $tType] :
( transp @ A
@ ^ [X2: A,Y3: A] : $false ) ).
% transp_empty
thf(fact_5660_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% transp_gr
thf(fact_5661_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_5662_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% transp_ge
thf(fact_5663_pred__prod__split,axiom,
! [B: $tType,A: $tType,P: $o > $o,Q2: A > $o,R4: B > $o,Xy2: product_prod @ A @ B] :
( ( P @ ( basic_pred_prod @ A @ B @ Q2 @ R4 @ Xy2 ) )
= ( ! [X2: A,Y3: B] :
( ( Xy2
= ( product_Pair @ A @ B @ X2 @ Y3 ) )
=> ( P
@ ( ( Q2 @ X2 )
& ( R4 @ Y3 ) ) ) ) ) ) ).
% pred_prod_split
thf(fact_5664_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P1: A > $o,A4: A,P25: B > $o,B3: B] :
( ( P1 @ A4 )
=> ( ( P25 @ B3 )
=> ( basic_pred_prod @ A @ B @ P1 @ P25 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ) ).
% pred_prod.intros
thf(fact_5665_pred__prod_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P15: A > $o,P26: B > $o,A8: product_prod @ A @ B] :
? [B6: A,C5: B] :
( ( A8
= ( product_Pair @ A @ B @ B6 @ C5 ) )
& ( P15 @ B6 )
& ( P26 @ C5 ) ) ) ) ).
% pred_prod.simps
thf(fact_5666_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P1: A > $o,P25: B > $o,A4: product_prod @ A @ B] :
( ( basic_pred_prod @ A @ B @ P1 @ P25 @ A4 )
=> ~ ! [A6: A,B2: B] :
( ( A4
= ( product_Pair @ A @ B @ A6 @ B2 ) )
=> ( ( P1 @ A6 )
=> ~ ( P25 @ B2 ) ) ) ) ).
% pred_prod.cases
thf(fact_5667_transp__inf,axiom,
! [A: $tType,R2: A > A > $o,S2: A > A > $o] :
( ( transp @ A @ R2 )
=> ( ( transp @ A @ S2 )
=> ( transp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S2 ) ) ) ) ).
% transp_inf
thf(fact_5668_transp__relcompp,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] : ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ R5 @ R5 ) @ R5 ) ) ) ).
% transp_relcompp
thf(fact_5669_transp__relcompp__less__eq,axiom,
! [A: $tType,R2: A > A > $o] :
( ( transp @ A @ R2 )
=> ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ R2 @ R2 ) @ R2 ) ) ).
% transp_relcompp_less_eq
thf(fact_5670_transp__trans__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X2: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% transp_trans_eq
thf(fact_5671_transp__trans,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] : ( trans @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ).
% transp_trans
thf(fact_5672_times__int__code_I4_J,axiom,
! [M: num,N2: num] :
( ( times_times @ int @ ( pos @ M ) @ ( neg @ N2 ) )
= ( neg @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_int_code(4)
thf(fact_5673_times__int__code_I5_J,axiom,
! [M: num,N2: num] :
( ( times_times @ int @ ( neg @ M ) @ ( pos @ N2 ) )
= ( neg @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_int_code(5)
thf(fact_5674_times__int__code_I6_J,axiom,
! [M: num,N2: num] :
( ( times_times @ int @ ( neg @ M ) @ ( neg @ N2 ) )
= ( pos @ ( times_times @ num @ M @ N2 ) ) ) ).
% times_int_code(6)
thf(fact_5675_repeat__mset__def,axiom,
! [A: $tType] :
( ( repeat_mset @ A )
= ( map_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ nat ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [N4: nat,M5: A > nat,A8: A] : ( times_times @ nat @ N4 @ ( M5 @ A8 ) ) ) ) ).
% repeat_mset_def
thf(fact_5676_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_5677_Nitpick_OEx1__unfold,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P2: A > $o] :
? [X2: A] :
( ( collect @ A @ P2 )
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Nitpick.Ex1_unfold
thf(fact_5678_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X1: A] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X1 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(3)
thf(fact_5679_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X22: B] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(4)
thf(fact_5680_asymp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A
@ ^ [X2: A,Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% asymp_greater
thf(fact_5681_Union__plus,axiom,
! [A: $tType,B: $tType,C: $tType,F2: ( sum_sum @ B @ C ) > ( set @ A ),A3: set @ B,B5: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( sum_sum @ B @ C ) @ ( set @ A ) @ F2 @ ( sum_Plus @ B @ C @ A3 @ B5 ) ) )
= ( sup_sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [A8: B] : ( F2 @ ( sum_Inl @ B @ C @ A8 ) )
@ A3 ) )
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [B6: C] : ( F2 @ ( sum_Inr @ C @ B @ B6 ) )
@ B5 ) ) ) ) ).
% Union_plus
thf(fact_5682_Union__sum,axiom,
! [C: $tType,A: $tType,B: $tType,F2: ( sum_sum @ A @ B ) > ( set @ C )] :
( ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ ( sum_sum @ A @ B ) @ ( set @ C ) @ F2 @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) )
= ( sup_sup @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [L2: A] : ( F2 @ ( sum_Inl @ A @ B @ L2 ) )
@ ( top_top @ ( set @ A ) ) ) )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ B @ ( set @ C )
@ ^ [R5: B] : ( F2 @ ( sum_Inr @ B @ A @ R5 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% Union_sum
thf(fact_5683_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A3 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A3 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B5 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_5684_case__sum__o__inj_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F2 @ G ) @ ( sum_Inl @ A @ C ) )
= F2 ) ).
% case_sum_o_inj(1)
thf(fact_5685_case__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F2 @ G ) @ ( sum_Inr @ C @ A ) )
= G ) ).
% case_sum_o_inj(2)
thf(fact_5686_asympD,axiom,
! [A: $tType,R4: A > A > $o,X: A,Y: A] :
( ( asymp @ A @ R4 )
=> ( ( R4 @ X @ Y )
=> ~ ( R4 @ Y @ X ) ) ) ).
% asympD
thf(fact_5687_asymp_Ointros,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [A6: A,B2: A] :
( ( R4 @ A6 @ B2 )
=> ~ ( R4 @ B2 @ A6 ) )
=> ( asymp @ A @ R4 ) ) ).
% asymp.intros
thf(fact_5688_asymp_Osimps,axiom,
! [A: $tType] :
( ( asymp @ A )
= ( ^ [A8: A > A > $o] :
? [R3: A > A > $o] :
( ( A8 = R3 )
& ! [X2: A,Y3: A] :
( ( R3 @ X2 @ Y3 )
=> ~ ( R3 @ Y3 @ X2 ) ) ) ) ) ).
% asymp.simps
thf(fact_5689_asymp_Ocases,axiom,
! [A: $tType,A4: A > A > $o] :
( ( asymp @ A @ A4 )
=> ! [A12: A,B15: A] :
( ( A4 @ A12 @ B15 )
=> ~ ( A4 @ B15 @ A12 ) ) ) ).
% asymp.cases
thf(fact_5690_Basic__BNF__LFPs_OInl__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( sum_Inl @ A @ B )
= ( ^ [A8: A] : ( basic_BNF_xtor @ ( sum_sum @ A @ B ) @ ( bNF_id_bnf @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A8 ) ) ) ) ) ).
% Basic_BNF_LFPs.Inl_def_alt
thf(fact_5691_Basic__BNF__LFPs_OInr__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( sum_Inr @ A @ B )
= ( ^ [A8: A] : ( basic_BNF_xtor @ ( sum_sum @ B @ A ) @ ( bNF_id_bnf @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ A8 ) ) ) ) ) ).
% Basic_BNF_LFPs.Inr_def_alt
thf(fact_5692_asymp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A @ ( ord_less @ A ) ) ) ).
% asymp_less
thf(fact_5693_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa: A > nat,X: B > nat,X1: A] :
( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inl @ A @ B @ X1 ) )
= ( plus_plus @ nat @ ( Xa @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(1)
thf(fact_5694_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa: A > nat,X: B > nat,X22: B] :
( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inr @ B @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(2)
thf(fact_5695_card__order__csum__cone__cexp__def,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A13: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ( bNF_Cardinal_cexp @ ( sum_sum @ B @ product_unit ) @ A @ ( bNF_Cardinal_csum @ B @ product_unit @ ( bNF_Ca6860139660246222851ard_of @ B @ A13 ) @ bNF_Cardinal_cone ) @ R2 )
= ( 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 ) @ A13 ) @ ( 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_5696_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_5697_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_5698_asymp__asym__eq,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( asymp @ A
@ ^ [A8: A,B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B6 ) @ R4 ) )
= ( asym @ A @ R4 ) ) ).
% asymp_asym_eq
thf(fact_5699_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_5700_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_5701_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R3 ) ) ) ) ).
% asym_iff
thf(fact_5702_asymD,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( asym @ A @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R4 ) ) ) ).
% asymD
thf(fact_5703_asym_Ointros,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ! [A6: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B2 ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A6 ) @ R4 ) )
=> ( asym @ A @ R4 ) ) ).
% asym.intros
thf(fact_5704_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A8: set @ ( product_prod @ A @ A )] :
? [R3: set @ ( product_prod @ A @ A )] :
( ( A8 = R3 )
& ! [X2: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R3 ) ) ) ) ) ).
% asym.simps
thf(fact_5705_asym_Ocases,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A4 )
=> ! [A12: A,B15: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B15 ) @ A4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B15 @ A12 ) @ A4 ) ) ) ).
% asym.cases
thf(fact_5706_asym__inv__image,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),F2: B > A] :
( ( asym @ A @ R4 )
=> ( asym @ B @ ( inv_image @ A @ B @ R4 @ F2 ) ) ) ).
% asym_inv_image
thf(fact_5707_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 ) ) )
@ ^ [B6: B] : ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_defs(1)
thf(fact_5708_sum__set__defs_I2_J,axiom,
! [C: $tType,D: $tType] :
( ( basic_setr @ C @ D )
= ( sum_case_sum @ C @ ( set @ D ) @ D
@ ^ [A8: C] : ( bot_bot @ ( set @ D ) )
@ ^ [Z3: D] : ( insert2 @ D @ Z3 @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% sum_set_defs(2)
thf(fact_5709_sum_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ nat @ ( sum_sum @ A @ B ) @ ( basic_BNF_size_sum @ C @ D @ F2 @ Fa ) @ ( sum_map_sum @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_sum @ A @ B @ ( comp @ C @ nat @ A @ F2 @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% sum.size_gen_o_map
thf(fact_5710_times__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( times_times @ nat ) @ ( times_times @ code_natural ) ).
% times_natural.transfer
thf(fact_5711_adm__wf__def,axiom,
! [B: $tType,A: $tType] :
( ( adm_wf @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
! [F4: A > B,G4: A > B,X2: A] :
( ! [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X2 ) @ R3 )
=> ( ( F4 @ Z3 )
= ( G4 @ Z3 ) ) )
=> ( ( F7 @ F4 @ X2 )
= ( F7 @ G4 @ X2 ) ) ) ) ) ).
% adm_wf_def
thf(fact_5712_map__sum__Inl__conv,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Fl: C > A,Fr: D > B,S2: sum_sum @ C @ D,Y: A] :
( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S2 )
= ( sum_Inl @ A @ B @ Y ) )
= ( ? [X2: C] :
( ( S2
= ( sum_Inl @ C @ D @ X2 ) )
& ( Y
= ( Fl @ X2 ) ) ) ) ) ).
% map_sum_Inl_conv
thf(fact_5713_map__sum__Inr__conv,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Fl: C > A,Fr: D > B,S2: sum_sum @ C @ D,Y: B] :
( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S2 )
= ( sum_Inr @ B @ A @ Y ) )
= ( ? [X2: D] :
( ( S2
= ( sum_Inr @ D @ C @ X2 ) )
& ( Y
= ( Fr @ X2 ) ) ) ) ) ).
% map_sum_Inr_conv
thf(fact_5714_map__sum__o__inj_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: A > B,G: D > C] :
( ( comp @ ( sum_sum @ A @ D ) @ ( sum_sum @ B @ C ) @ A @ ( sum_map_sum @ A @ B @ D @ C @ F2 @ G ) @ ( sum_Inl @ A @ D ) )
= ( comp @ B @ ( sum_sum @ B @ C ) @ A @ ( sum_Inl @ B @ C ) @ F2 ) ) ).
% map_sum_o_inj(1)
thf(fact_5715_map__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: A > B,G: D > C] :
( ( comp @ ( sum_sum @ A @ D ) @ ( sum_sum @ B @ C ) @ D @ ( sum_map_sum @ A @ B @ D @ C @ F2 @ G ) @ ( sum_Inr @ D @ A ) )
= ( comp @ C @ ( sum_sum @ B @ C ) @ D @ ( sum_Inr @ C @ B ) @ G ) ) ).
% map_sum_o_inj(2)
thf(fact_5716_sum_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R13: A > C > $o,R24: B > D > $o] :
( relcompp @ ( sum_sum @ A @ B ) @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( conversep @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R13 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R24 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R13 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R24 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% sum.rel_compp_Grp
thf(fact_5717_sum_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R13: A > C > $o,R24: B > D > $o,A8: sum_sum @ A @ B,B6: sum_sum @ C @ D] :
? [Z3: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z3
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R13 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R24 ) ) ) ) ) )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z3 )
= A8 )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z3 )
= B6 ) ) ) ) ).
% sum.in_rel
thf(fact_5718_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A,S: set @ B,T3: B > A > $o] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ S )
=> ( ( T3
= ( ^ [X2: B,Y3: A] :
( X2
= ( Rep2 @ Y3 ) ) ) )
=> ( quotient @ B @ A
@ ( bNF_eq_onp @ B
@ ^ [X2: B] : ( member @ B @ X2 @ S ) )
@ Abs2
@ Rep2
@ T3 ) ) ) ).
% typedef_to_Quotient
thf(fact_5719_rel__sum_Ointros_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R12: A > C > $o,A4: A,C2: C,R23: B > D > $o] :
( ( R12 @ A4 @ C2 )
=> ( bNF_rel_sum @ A @ C @ B @ D @ R12 @ R23 @ ( sum_Inl @ A @ B @ A4 ) @ ( sum_Inl @ C @ D @ C2 ) ) ) ).
% rel_sum.intros(1)
thf(fact_5720_rel__sum_Ointros_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R23: B > D > $o,B3: B,D3: D,R12: A > C > $o] :
( ( R23 @ B3 @ D3 )
=> ( bNF_rel_sum @ A @ C @ B @ D @ R12 @ R23 @ ( sum_Inr @ B @ A @ B3 ) @ ( sum_Inr @ D @ C @ D3 ) ) ) ).
% rel_sum.intros(2)
thf(fact_5721_Quotient__filter,axiom,
! [B: $tType,A: $tType,R4: A > A > $o,Abs2: A > B,Rep2: B > A,T3: A > B > $o] :
( ( quotient @ A @ B @ R4 @ Abs2 @ Rep2 @ T3 )
=> ( quotient @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ A @ R4 ) @ ( filtermap @ A @ B @ Abs2 ) @ ( filtermap @ B @ A @ Rep2 ) @ ( rel_filter @ A @ B @ T3 ) ) ) ).
% Quotient_filter
thf(fact_5722_rel__sum_Osimps,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R13: A > C > $o,R24: B > D > $o,A15: sum_sum @ A @ B,A24: sum_sum @ C @ D] :
( ? [A8: A,C5: C] :
( ( A15
= ( sum_Inl @ A @ B @ A8 ) )
& ( A24
= ( sum_Inl @ C @ D @ C5 ) )
& ( R13 @ A8 @ C5 ) )
| ? [B6: B,D5: D] :
( ( A15
= ( sum_Inr @ B @ A @ B6 ) )
& ( A24
= ( sum_Inr @ D @ C @ D5 ) )
& ( R24 @ B6 @ D5 ) ) ) ) ) ).
% rel_sum.simps
thf(fact_5723_rel__sum_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > C > $o,R23: B > D > $o,A1: sum_sum @ A @ B,A22: sum_sum @ C @ D] :
( ( bNF_rel_sum @ A @ C @ B @ D @ R12 @ R23 @ A1 @ A22 )
=> ( ! [A6: A] :
( ( A1
= ( sum_Inl @ A @ B @ A6 ) )
=> ! [C3: C] :
( ( A22
= ( sum_Inl @ C @ D @ C3 ) )
=> ~ ( R12 @ A6 @ C3 ) ) )
=> ~ ! [B2: B] :
( ( A1
= ( sum_Inr @ B @ A @ B2 ) )
=> ! [D2: D] :
( ( A22
= ( sum_Inr @ D @ C @ D2 ) )
=> ~ ( R23 @ B2 @ D2 ) ) ) ) ) ).
% rel_sum.cases
thf(fact_5724_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A,T3: B > A > $o] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( top_top @ ( set @ B ) ) )
=> ( ( T3
= ( ^ [X2: B,Y3: A] :
( X2
= ( Rep2 @ Y3 ) ) ) )
=> ( quotient @ B @ A
@ ^ [Y4: B,Z5: B] : Y4 = Z5
@ Abs2
@ Rep2
@ T3 ) ) ) ).
% UNIV_typedef_to_Quotient
thf(fact_5725_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep2: A > B,Abs2: B > A,P: B > $o,T3: B > A > $o] :
( ( type_definition @ A @ B @ Rep2 @ Abs2 @ ( collect @ B @ P ) )
=> ( ( T3
= ( ^ [X2: B,Y3: A] :
( X2
= ( Rep2 @ Y3 ) ) ) )
=> ( quotient @ B @ A @ ( bNF_eq_onp @ B @ P ) @ Abs2 @ Rep2 @ T3 ) ) ) ).
% open_typedef_to_Quotient
thf(fact_5726_Quotient__eq__onp__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs2: A > B,Rep2: B > A,Cr: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs2 @ Rep2 @ Cr )
=> ( type_definition @ B @ A @ Rep2 @ Abs2 @ ( collect @ A @ P ) ) ) ).
% Quotient_eq_onp_typedef
thf(fact_5727_Quotient__eq__onp__type__copy,axiom,
! [B: $tType,A: $tType,Abs2: A > B,Rep2: B > A,Cr: A > B > $o] :
( ( quotient @ A @ B
@ ^ [Y4: A,Z5: A] : Y4 = Z5
@ Abs2
@ Rep2
@ Cr )
=> ( type_definition @ B @ A @ Rep2 @ Abs2 @ ( top_top @ ( set @ A ) ) ) ) ).
% Quotient_eq_onp_type_copy
thf(fact_5728_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_finite2 @ 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_5729_semilattice__neutr__set_Oinsert,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( insert2 @ A @ X @ A3 ) )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A3 ) ) ) ) ) ).
% semilattice_neutr_set.insert
thf(fact_5730_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( F2 @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_neutr_set.closed
thf(fact_5731_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_5732_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite2 @ 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_5733_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P3: B > $o,F2: C > B] :
( ( nO_MATCH @ ( A > A ) @ ( B > $o ) @ ( id @ A ) @ P3 )
=> ( ( basic_pred_fun @ C @ B
@ ^ [X2: C] : $true
@ P3
@ F2 )
= ( basic_pred_fun @ C @ $o
@ ^ [X2: C] : $true
@ ( id @ $o )
@ ( comp @ B @ $o @ C @ P3 @ F2 ) ) ) ) ).
% pred_fun_True_id
thf(fact_5734_Fpow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_Fpow @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_5735_times__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( times_times @ nat @ Xa @ X ) ) ) ).
% times_natural.abs_eq
thf(fact_5736_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_5737_type__definition__natural,axiom,
type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% type_definition_natural
thf(fact_5738_times__natural__def,axiom,
( ( times_times @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( times_times @ nat ) ) ) ).
% times_natural_def
thf(fact_5739_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N2: multiset @ A,M: multiset @ A] :
( ( subseteq_mset @ A @ N2 @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 )
= N2 ) ) ).
% subset_mset.min_arg_le(1)
thf(fact_5740_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M: multiset @ A,N2: multiset @ A] :
( ( subseteq_mset @ A @ M @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 )
= M ) ) ).
% subset_mset.min_arg_le(2)
thf(fact_5741_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A8: A,B6: A] : ( if @ A @ ( Less_eq2 @ A8 @ B6 ) @ A8 @ B6 ) ) ) ).
% ord.min_def
thf(fact_5742_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_5743_prop__match,axiom,
! [A: $tType,P: A > $o,Al: list @ A,E4: A,E5: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E4 )
=> ( ~ ( P @ E5 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E4 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E5 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( E4 = E5 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
thf(fact_5744_prop__matchD,axiom,
! [A: $tType,Al: list @ A,E4: A,Bl: list @ A,Al2: list @ A,E5: A,Bl2: list @ A,P: A > $o] :
( ( ( append @ A @ Al @ ( cons @ A @ E4 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E5 @ Bl2 ) ) )
=> ( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E4 )
=> ( ~ ( P @ E5 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( Al = Al2 )
& ( E4 = E5 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
thf(fact_5745_times__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( times_times @ int ) @ ( times_times @ code_integer ) ).
% times_integer.transfer
thf(fact_5746_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ 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_5747_semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ 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_5748_times__integer__def,axiom,
( ( times_times @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( times_times @ int ) ) ) ).
% times_integer_def
thf(fact_5749_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert2 @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ F2 @ X @ A3 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_5750_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_5751_semilattice__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( F2 @ X3 @ Y2 ) @ ( insert2 @ A @ X3 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_5752_semilattice__set_Oinsert,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ 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_5753_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ 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_5754_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F2: A > A > A,H2: A > A,N: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ! [X3: A,Y2: A] :
( ( H2 @ ( F2 @ X3 @ Y2 ) )
= ( F2 @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ N ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ ( image2 @ A @ A @ H2 @ N ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_5755_semilattice__set_Osubset,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,B5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ B5 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_5756_semilattice__set_Ounion,axiom,
! [A: $tType,F2: A > A > A,A3: set @ A,B5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ B5 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_5757_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,B5: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F2 @ B5 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_5758_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_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ A3 ) )
=> ! [A12: A] :
( ( member @ A @ A12 @ A3 )
=> ( Less_eq @ X @ A12 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_5759_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_finite2 @ 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_5760_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_finite2 @ 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
% Type constructors (630)
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_1,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_2,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice_top @ ( option @ A20 ) ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_3,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( filter @ A20 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_4,axiom,
bounded_lattice_top @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_5,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( set @ A20 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top_6,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( condit1219197933456340205attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple592849572758109894attice @ A21 )
=> ( comple592849572758109894attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounde4967611905675639751up_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple6319245703460814977attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A20: $tType,A21: $tType] :
( ( boolea8198339166811842893lgebra @ A21 )
=> ( boolea8198339166811842893lgebra @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple9053668089753744459l_ccpo @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_sup @ A21 )
=> ( semilattice_sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_inf @ A21 )
=> ( semilattice_inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( distrib_lattice @ A21 )
=> ( distrib_lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A20: $tType,A21: $tType] :
( ( order_top @ A21 )
=> ( order_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( order_bot @ A21 )
=> ( order_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A20: $tType,A21: $tType] :
( ( preorder @ A21 )
=> ( preorder @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A20: $tType,A21: $tType] :
( ( lattice @ A21 )
=> ( lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A20: $tType,A21: $tType] :
( ( order @ A21 )
=> ( order @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A20: $tType,A21: $tType] :
( ( top @ A21 )
=> ( top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A20: $tType,A21: $tType] :
( ( ord @ A21 )
=> ( ord @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A20: $tType,A21: $tType] :
( ( bot @ A21 )
=> ( bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A20: $tType,A21: $tType] :
( ( uminus @ A21 )
=> ( uminus @ ( A20 > A21 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_7,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___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___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___Lattices_Osemilattice__sup_8,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_9,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_10,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_11,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_12,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_13,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_14,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_15,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ 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_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_16,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_17,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_18,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_19,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_20,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_21,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_22,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_23,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_24,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_25,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_26,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_27,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_28,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_29,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_30,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_31,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_32,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_33,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_34,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_35,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_36,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_37,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_38,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_39,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_40,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_41,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_47,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_48,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_49,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_50,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_51,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_52,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_53,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_54,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_55,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_56,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_57,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_58,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_59,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_60,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_61,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_62,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_63,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_64,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_65,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_73,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_74,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_75,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_76,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_77,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_78,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_79,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_80,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_81,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_82,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_83,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_84,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_85,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_86,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_87,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_88,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_89,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_90,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_91,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Power_Opower_92,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_93,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_94,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_95,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_96,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_97,axiom,
heap @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_98,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_99,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_100,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_101,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_102,axiom,
times @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_103,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_104,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_105,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_106,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_107,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_108,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_109,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_110,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_111,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_112,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_113,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_114,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_115,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_116,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_117,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_118,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_119,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_120,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_121,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_122,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_123,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_124,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_125,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_126,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_127,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_128,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_129,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_130,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_131,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_132,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_133,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_134,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_135,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_136,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_137,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_138,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_139,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_140,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_141,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_142,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_143,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_144,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_145,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_146,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_147,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_148,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_149,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_150,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_151,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_152,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_153,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_154,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_155,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_156,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_157,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_158,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_159,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_160,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_161,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_162,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_163,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_164,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_165,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_166,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_167,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_168,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_169,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_170,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_171,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_172,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_173,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_174,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_175,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_176,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_177,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_178,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_179,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_180,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_181,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_182,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_183,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_184,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_185,axiom,
! [A20: $tType] : ( condit1219197933456340205attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_186,axiom,
! [A20: $tType] : ( comple592849572758109894attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_187,axiom,
! [A20: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_188,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_189,axiom,
! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_190,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_191,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_192,axiom,
! [A20: $tType] : ( semilattice_sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_193,axiom,
! [A20: $tType] : ( semilattice_inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_194,axiom,
! [A20: $tType] : ( distrib_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_195,axiom,
! [A20: $tType] : ( bounded_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_196,axiom,
! [A20: $tType] : ( order_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_197,axiom,
! [A20: $tType] : ( order_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_198,axiom,
! [A20: $tType] : ( preorder @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_199,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_200,axiom,
! [A20: $tType] : ( lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_201,axiom,
! [A20: $tType] : ( order @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_202,axiom,
! [A20: $tType] : ( top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_203,axiom,
! [A20: $tType] : ( ord @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_204,axiom,
! [A20: $tType] : ( bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_205,axiom,
! [A20: $tType] : ( uminus @ ( set @ A20 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_206,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_207,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_208,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_209,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_210,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_211,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_212,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_213,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_214,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_215,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_216,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_217,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_218,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_219,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_220,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_221,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_222,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_223,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_224,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_225,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_226,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_227,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_228,axiom,
heap @ $o ).
thf(tcon_Heap_Oref___Heap_Oheap_229,axiom,
! [A20: $tType] : ( heap @ ( ref @ A20 ) ) ).
thf(tcon_List_Olist___Heap_Oheap_230,axiom,
! [A20: $tType] :
( ( heap @ A20 )
=> ( heap @ ( list @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_231,axiom,
! [A20: $tType] : ( heap @ ( array @ A20 ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_232,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_233,axiom,
! [A20: $tType,A21: $tType] :
( ( ( heap @ A20 )
& ( heap @ A21 ) )
=> ( heap @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_234,axiom,
! [A20: $tType] : ( condit1219197933456340205attice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_235,axiom,
! [A20: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_236,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_237,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_238,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_239,axiom,
! [A20: $tType] : ( semilattice_sup @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_240,axiom,
! [A20: $tType] : ( semilattice_inf @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_241,axiom,
! [A20: $tType] : ( distrib_lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_242,axiom,
! [A20: $tType] : ( bounded_lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_243,axiom,
! [A20: $tType] : ( order_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_244,axiom,
! [A20: $tType] : ( order_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_245,axiom,
! [A20: $tType] : ( preorder @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_246,axiom,
! [A20: $tType] : ( lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_247,axiom,
! [A20: $tType] : ( order @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_248,axiom,
! [A20: $tType] : ( top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_249,axiom,
! [A20: $tType] : ( ord @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_250,axiom,
! [A20: $tType] : ( bot @ ( filter @ A20 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_251,axiom,
! [A20: $tType] :
( ( comple5582772986160207858norder @ A20 )
=> ( condit6923001295902523014norder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_252,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_253,axiom,
! [A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_254,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A20: $tType] :
( ( comple5582772986160207858norder @ A20 )
=> ( comple5582772986160207858norder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_255,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_256,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( bounded_lattice_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_257,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_258,axiom,
! [A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_259,axiom,
! [A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_260,axiom,
! [A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_261,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_262,axiom,
! [A20: $tType] :
( ( wellorder @ A20 )
=> ( wellorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_263,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_264,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_265,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_266,axiom,
! [A20: $tType] :
( ( linorder @ A20 )
=> ( linorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_267,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_268,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_269,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_270,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_271,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_272,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_273,axiom,
! [A20: $tType] :
( ( heap @ A20 )
=> ( heap @ ( option @ A20 ) ) ) ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_274,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_275,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_276,axiom,
bounded_lattice_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_277,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_278,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_279,axiom,
distrib_lattice @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_280,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_281,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_282,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_283,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_284,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_285,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_286,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_287,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_288,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_289,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Otop_290,axiom,
top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_291,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_292,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_293,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_294,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_295,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_296,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_297,axiom,
dvd @ assn ).
thf(tcon_Typerep_Otyperep___Heap_Oheap_298,axiom,
heap @ typerep ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_299,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_300,axiom,
! [A20: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_301,axiom,
! [A20: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_302,axiom,
! [A20: $tType] : ( cancel_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_303,axiom,
! [A20: $tType] : ( comm_monoid_diff @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_304,axiom,
! [A20: $tType] : ( ab_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_305,axiom,
! [A20: $tType] : ( comm_monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_306,axiom,
! [A20: $tType] : ( semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_307,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_308,axiom,
! [A20: $tType] : ( monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_309,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( order @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_310,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_311,axiom,
! [A20: $tType] : ( zero @ ( multiset @ A20 ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_312,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_313,axiom,
! [A20: $tType,A21: $tType] :
( ( ( heap @ A20 )
& ( heap @ A21 ) )
=> ( heap @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_314,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_315,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_316,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_317,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_318,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_319,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_320,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_321,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_322,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_323,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_324,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_325,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_326,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_327,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_328,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_329,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_330,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_331,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_332,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_333,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_334,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_335,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_336,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_337,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_338,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_339,axiom,
heap @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_340,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_341,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_342,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_343,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_344,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_345,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_346,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_347,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_348,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_349,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_350,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_351,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_352,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_353,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_354,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_355,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_356,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_357,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_358,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_359,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_360,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_361,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_362,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_363,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_364,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_365,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_366,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_367,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_368,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_369,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_370,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_371,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_372,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_373,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_374,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_375,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_376,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_377,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_378,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_379,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_380,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_381,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_382,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_383,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_384,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_385,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_386,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_387,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_388,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_389,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_390,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_391,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_392,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_393,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_394,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_395,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_396,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_397,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_398,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_399,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_400,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_401,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_402,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_403,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_404,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_405,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_406,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_407,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_408,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_409,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_410,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_411,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_412,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_413,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_420,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_421,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_422,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_423,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_424,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_425,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_426,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_427,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_428,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_429,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_430,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_431,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_432,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_433,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_434,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_435,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_436,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_437,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_438,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_439,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_440,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_441,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_442,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_443,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_444,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_445,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_446,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_447,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_448,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_449,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_450,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_451,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_452,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_453,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_454,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_455,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_456,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_457,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_458,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_459,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_460,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_461,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_462,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_463,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_464,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_465,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_466,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_467,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_468,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_469,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_470,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_471,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_472,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_473,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_474,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_475,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_476,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_477,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_478,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_479,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_480,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_481,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_482,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_483,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_484,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_485,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_486,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_487,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_488,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_489,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_490,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_491,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_492,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_493,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_494,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_495,axiom,
dvd @ code_natural ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
%------------------------------------------------------------------------------